<?xml version="1.0"?> 
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "mathml.dtd"> 
<?xml-stylesheet type="text/css" href="testmath.css"?> 
<html  
xmlns="http://www.w3.org/1999/xhtml"  
><head>
   <title>Sample Paper for the amsmath Package File name: testmath.tex</title> 
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" /> 
<meta name="generator" content="TeX4ht (http://www.cis.ohio-state.edu/~gurari/TeX4ht/mn.html)" /> 
<!-- early_,early^,xhtml,mozilla --> 
<meta name="src" content="testmath.tex" /> 
<meta name="date" content="2002-02-26 12:35:00" /> 
<link rel="stylesheet" type="text/css" href="testmath.css" /> 
</head><body 
>
   <div align="center" class="maketitle">

<h2 class="titleHead">Sample Paper for the amsmath Package<br />
File name: testmath.tex</h2>
<div class="author" align="center"><span 
class="cmr-12">American Mathematical Society</span></div>
<br />
<div class="date" align="center"><span 
class="cmr-12">Version 2.0, 1999/11/15</span></div>
   </div>
   <h3 class="sectionHead"><span class="titlemark">1</span> <a 
  name="x1-10001"></a>Introduction</h3>
<!--l. 147--><p class="noindent">This paper contains examples of various features from <span 
class="cmsy-10"><!--span 
class="htf-calligraphy"-->A<!--/span--></span><span 
class="cmsy-10"><!--span 
class="htf-calligraphy"-->M<!--/span--></span> <span 
class="cmsy-10"><!--span 
class="htf-calligraphy"-->S<!--/span--></span>-<span class="LATEX">L<span class="A">A</span><span class="TEX">T<span 
class="E">E</span>X</span></span>.
</p>
   <h3 class="sectionHead"><span class="titlemark">2</span> <a 
  name="x1-20002"></a>Enumeration of Hamiltonian paths in a graph</h3>
<!--l. 151--><p class="noindent">Let <!--l. 151--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>A</mi> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math> be the adjacency matrix
of graph <!--l. 151--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>G</mi></math>. The corresponding
Kirchhoff matrix <!--l. 152--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>K</mi> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>k</mi></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
is obtained from <!--l. 153--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>A</mi></math>
by replacing in <!--l. 153--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mo 
class="MathClass-bin">&#x2212;</mo><mi 
>A</mi></math>
each diagonal entry by the degree of its corresponding vertex; i.e., the <!--l. 154--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>i</mi></math>th
diagonal entry is identified with the degree of the <!--l. 155--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>i</mi></math>th
vertex. It is well known that </p><table class="equation"><tr><td>

<!--l. 159--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<mo 
>det</mo><mi 
>K</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>i</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>i</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <!--mstyle 
class="text"--><mtext >&#x000A0;the&#x000A0;number&#x000A0;of&#x000A0;spanning&#x000A0;trees&#x000A0;of&#x000A0;</mtext><!--mstyle 
class="math"--><mi 
>G</mi><!--/mstyle--><mtext ></mtext><!--/mstyle--><mo 
class="MathClass-punc">,</mo><mi 
>i</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><mo>&#x2026;</mo><mo 
class="MathClass-punc">,</mo><mi 
>n</mi><mstyle 
    class="label" id="x1-2001r1"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(1)</td></tr></table>
where <!--l. 160--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>K</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>i</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>i</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math> is the <!--l. 160--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>i</mi></math>th principal
submatrix of <!--l. 161--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>K</mi></math>.

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;det&#x005C;mathbf{K}(i|i)=&#x005C;text{&#x00A0;the&#x00A0;number&#x00A0;of&#x00A0;spanning&#x00A0;trees&#x00A0;of&#x00A0;$G$},
</div>
</td></tr></table>
<!--l. 166--><p class="indent">   Let <!--l. 166--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>C</mi></mrow><mrow 
><mi 
>i</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>j</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
></math> be the set of
graphs obtained from <!--l. 166--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>G</mi></math>
by attaching edge <!--l. 167--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><msub><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math> to
each spanning tree of <!--l. 167--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>G</mi></math>.
Denote by <!--l. 167--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>C</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x22C3;</mo>
  </mrow><mrow 
><mi 
>j</mi></mrow></msub 
><msub><mrow 
><mi 
>C</mi></mrow><mrow 
><mi 
>i</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>j</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
></math>.
It is obvious that the collection of Hamiltonian cycles is a subset of <!--l. 169--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>C</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></math>. Note that the
cardinality of <!--l. 169--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>C</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></math>
is <!--l. 169--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>k</mi></mrow><mrow 
><mi 
>i</mi><mi 
>i</mi></mrow></msub 
><mo 
> det</mo><mi 
>K</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>i</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>i</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>.
Let <!--l. 170--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mover 
accent="false"><mrow 
><mi 
>X</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">{</mo><mrow><msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">}</mo></mrow></math>.

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
$&#x005C;wh&#x00A0;X=&#x005C;{&#x005C;hat&#x00A0;x&#x005F;1,&#x005C;dots,&#x005C;hat&#x00A0;x&#x005F;n&#x005C;}$
</div>
</td></tr></table>
<!--l. 174--><p class="indent">   Define multiplication for the elements of <!--l. 174--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mover 
accent="false"><mrow 
><mi 
>X</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></math> by
</p><table class="equation"><tr><td>
<!--l. 178--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><msubsup><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>i</mi></mrow><mrow 
><mn>2</mn></mrow></msubsup 
> <mo 
class="MathClass-rel">=</mo> <mn>0</mn><mo 
class="MathClass-punc">,</mo><mi 
>i</mi><mo 
class="MathClass-punc">,</mo><mi 
>j</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><mi 
>n</mi><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-2002r2"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(2)</td></tr></table>
Let <!--l. 179--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>k</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>k</mi></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow></msub 
><msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>j</mi></mrow></msub 
></math> and <!--l. 179--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>k</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
  </mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">/</mo><mo 
class="MathClass-rel">=</mo><mi 
>i</mi></mrow></msub 
><msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>k</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow></msub 
></math>. Then the number of
Hamiltonian cycles <!--l. 180--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
></math>
is given by the relation <span class="cite">[<a 
href="#Xliuchow:formalsum">8</a>]</span> <table class="equation"><tr><td>

<!--l. 185--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<mrow><mo 
class="MathClass-open">(</mo><mrow><msubsup><mrow 
><mo 
class="MathClass-op">&#x220F;</mo>
   </mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>n</mi></mrow></msubsup 
><msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
>
<mi 
>j</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><mn>2</mn></mrow></mfrac><msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>k</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow></msub 
><mo 
> det</mo><mover 
accent="false"><mrow 
><mi 
>K</mi></mrow><mo>&#x0302;</mo></mover><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>i</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>i</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo><mspace width="2em" class="qquad"/><mi 
>i</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><mo>&#x2026;</mo><mo 
class="MathClass-punc">,</mo><mi 
>n</mi><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-2003r3"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(3)</td></tr></table>
The task here is to express (<a 
href="#x1-2003r3">3<!--tex4ht:ref: H-cycles--></a>) in a form free of any <!--l. 187--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></math>, <!--l. 188--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>i</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><mi 
>n</mi></math>. The
result also leads to the resolution of enumeration of Hamiltonian paths in a
graph.
   It is well known that the enumeration of Hamiltonian cycles and paths in a complete graph <!--l. 192--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>K</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></math> and in a complete
bipartite graph <!--l. 192--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>K</mi></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mrow></msub 
></math>
can only be found from <span 
class="cmti-10">first combinatorial principles </span><span class="cite">[<a 
href="#Xhapa:graphenum">4</a>]</span>. One wonders if
there exists a formula which can be used very efficiently to produce <!--l. 195--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>K</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></math> and <!--l. 195--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>K</mi></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mrow></msub 
></math>.
Recently, using Lagrangian methods, Goulden and Jackson have shown that <!--l. 196--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
></math> can
be expressed in terms of the determinant and permanent of the adjacency
matrix <span class="cite">[<a 
href="#Xgouja:lagrmeth">3</a>]</span>. However, the formula of Goulden and Jackson determines neither <!--l. 199--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>K</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></math> nor <!--l. 199--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>K</mi></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mrow></msub 
></math> effectively. In
this paper, using an algebraic method, we parametrize the adjacency matrix. The resulting
formula also involves the determinant and permanent, but it can easily be applied to <!--l. 202--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>K</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></math> and <!--l. 202--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">

<msub><mrow 
><mi 
>K</mi></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mrow></msub 
></math>. In addition, we eliminate
the permanent from <!--l. 203--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
></math>
and show that <!--l. 203--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
></math>
can be represented by a determinantal function of multivariables, each variable with domain <!--l. 205--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mrow><mo 
class="MathClass-open">{</mo><mrow><mn>0</mn><mo 
class="MathClass-punc">,</mo><mn>1</mn></mrow><mo 
class="MathClass-close">}</mo></mrow></math>. Furthermore, we
show that <!--l. 205--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
></math>
can be written by number of spanning trees of subgraphs.
Finally, we apply the formulas to a complete multigraph <!--l. 207--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>K</mi></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-op">&#x2026;</mo><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>p</mi></mrow></msub 
></mrow></msub 
></math>.
<!--l. 209--><p class="indent">   The conditions <!--l. 209--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>j</mi><mi 
>i</mi></mrow></msub 
></math>,
<!--l. 209--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">     <mi 
>i</mi><mo 
class="MathClass-punc">,</mo><mi 
>j</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><mi 
>n</mi></math>, are
not required in this paper. All formulas can be extended to a digraph simply by multiplying <!--l. 211--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
></math> by
2.
</p>
   <h3 class="sectionHead"><span class="titlemark">3</span> <a 
  name="x1-30003"></a>Main Theorem</h3>
   <div class="newtheorem">
<!--l. 216--><p class="noindent"><span class="head">
<span 
class="cmti-10">Notation.</span> </span>For <!--l. 216--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>p</mi><mo 
class="MathClass-punc">,</mo><mi 
>q</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>P</mi></math>
and <!--l. 216--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>n</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>&#x03C9;</mi></math> we
write <!--l. 217--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>q</mi><mo 
class="MathClass-punc">,</mo><mi 
>n</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2264;</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>p</mi><mo 
class="MathClass-punc">,</mo><mi 
>n</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
if <!--l. 217--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>q</mi> <mo 
class="MathClass-rel">&#x2264;</mo> <mi 
>p</mi></math>
and <!--l. 217--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mi 
>q</mi><mo 
class="MathClass-punc">,</mo><mi 
>n</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mi 
>p</mi><mo 
class="MathClass-punc">,</mo><mi 
>n</mi></mrow></msub 
></math>.

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{notation}&#x00A0;For&#x00A0;$p,q&#x005C;in&#x00A0;P$&#x00A0;and&#x00A0;$n&#x005C;in&#x005C;omega$
&#x00A0;<br />...
&#x00A0;<br />&#x005C;end{notation}
</div>
</td></tr></table>
   </div>
<!--l. 225--><p class="indent">   Let <!--l. 225--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>B</mi> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math> be
an <!--l. 225--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>n</mi> <mo 
class="MathClass-bin">&#x00D7;</mo> <mi 
>n</mi></math> matrix.
Let <!--l. 225--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>n</mi> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">{</mo><mrow><mn>1</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><mi 
>n</mi></mrow><mo 
class="MathClass-close">}</mo></mrow></math>.
Using the properties of (<a 
href="#x1-2002r2">2<!--tex4ht:ref: multdef--></a>), it is readily seen that
</p>
   <div class="newtheorem">
<!--l. 229--><p class="noindent"><span class="head">
<a 
  name="x1-3001r1"></a>
<span 
class="cmbx-10">Lemma 3.1.</span> </span> </p><table class="equation"><tr><td>
<!--l. 234--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">      
<msub><mrow 
><mo 
class="MathClass-op">&#x220F;</mo>
  </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msub><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow></msub 
><msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mo 
class="MathClass-op">&#x220F;</mo>
   </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-op">per</mo><!--nolimits--><mi 
>B</mi><mstyle 
    class="label" id="x1-3002r4"  ></mstyle><!--endlabel-->
</math>

</td><td class="eq-no">(4)</td></tr></table>
<span 
class="cmti-10">where </span><!--l. 235--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mo 
class="MathClass-op">per</mo><!--nolimits--><mi 
>B</mi></math> <span 
class="cmti-10">is the</span>
<span 
class="cmti-10">permanent of </span><!--l. 235--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>B</mi></math><span 
class="cmti-10">.</span>
   </div>
<!--l. 238--><p class="indent">   Let <!--l. 238--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mover 
accent="false"><mrow 
><mi 
>Y</mi> </mrow><mo 
class="MathClass-op">&#x0302;</mo></mover> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">{</mo><mrow><msub><mrow 
><mi 
>&#x0177;</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>&#x0177;</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">}</mo></mrow></math>. Define multiplication
for the elements of <!--l. 239--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mover 
accent="false"><mrow 
><mi 
>Y</mi> </mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></math>
by </p><table class="equation"><tr><td>
<!--l. 242--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<msub><mrow 
><mi 
>&#x0177;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><msub><mrow 
><mi 
>&#x0177;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-bin">+</mo><msub><mrow 
> <mi 
>&#x0177;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><msub><mrow 
><mi 
>&#x0177;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mn>0</mn><mo 
class="MathClass-punc">,</mo><mi 
>i</mi><mo 
class="MathClass-punc">,</mo><mi 
>j</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><mi 
>n</mi><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-3003r5"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(5)</td></tr></table>
Then, it follows that <div class="newtheorem">
   <span class="head">
<a 
  name="x1-3004r2"></a>
<span 
class="cmbx-10">Lemma 3.2.</span> </span> <table class="equation"><tr><td>

<!--l. 249--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">      
<msub><mrow 
><mo 
class="MathClass-op">&#x220F;</mo>
  </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msub><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow></msub 
><msub><mrow 
><mi 
>&#x0177;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mo 
class="MathClass-op">&#x220F;</mo>
   </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msub><mrow 
><mi 
>&#x0177;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
>det</mo><mi 
>B</mi><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-3005r6"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(6)</td></tr></table>
   </div>
<!--l. 252--><p class="indent">   Note that all basic properties of determinants are direct consequences of Lemma
&#x00A0;<a 
href="#x1-3004r2">3.2<!--tex4ht:ref: lem-det--></a>. Write </p><table class="equation"><tr><td>
<!--l. 257--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<msub><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msub><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow></msub 
><msub><mrow 
><mi 
>&#x0177;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msubsup><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msubsup 
><msub><mrow 
><mi 
>&#x0177;</mi></mrow><mrow 
>
<mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>i</mi><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>&#x03BB;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>&#x0177;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mi 
>&#x0177;</mi><mstyle 
    class="label" id="x1-3006r7"  ></mstyle><!--endlabel-->
</math>

</td><td class="eq-no">(7)</td></tr></table>
where <table class="equation"><tr><td>
<!--l. 262--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<msubsup><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>i</mi><mi 
>i</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msubsup 
> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>&#x03BB;</mi></mrow><mrow 
>
<mi 
>i</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><msubsup><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msubsup 
> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>b</mi></mrow><mrow 
>
<mi 
>i</mi><mi 
>j</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mi 
>i</mi> <mo 
class="MathClass-rel">/</mo><mo 
class="MathClass-rel">=</mo> <mi 
>j</mi><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-3007r8"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(8)</td></tr></table>
Let <!--l. 263--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msup><mrow 
><mi 
>B</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><msubsup><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msubsup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>.
By (<a 
href="#x1-3005r6">6<!--tex4ht:ref: detprod--></a>) and (<a 
href="#x1-3006r7">7<!--tex4ht:ref: sum-bij--></a>), it is straightforward to show the following result: <div class="newtheorem">
   <span class="head">
<a 
  name="x1-3008r3"></a>
<span 
class="cmbx-10">Theorem 3.3.</span> </span> <table class="equation"><tr><td>

<!--l. 273--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">      
<mo 
>det</mo><mi 
>B</mi> <mo 
class="MathClass-rel">=</mo><msubsup><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
  </mrow><mrow 
><mi 
>l</mi><mo 
class="MathClass-rel">=</mo><mn>0</mn></mrow><mrow 
><mi 
>n</mi></mrow></msubsup 
><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
><mo 
class="MathClass-rel">&#x2286;</mo><mi 
>n</mi></mrow></msub 
><msub><mrow 
> <mo 
class="MathClass-op">&#x220F;</mo>
        </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2208;</mo><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>i</mi><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>&#x03BB;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
>det</mo><msup><mrow 
><mi 
>B</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>I</mi></mrow><mrow 
>
<mi 
>l</mi></mrow></msub 
><mo 
class="MathClass-rel">&#x2223;</mo><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo><mstyle 
    class="label" id="x1-3009r9"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(9)</td></tr></table>
<span 
class="cmti-10">where </span><!--l. 274--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">{</mo><mrow><msub><mrow 
><mi 
>i</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>i</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">}</mo></mrow></math> <span 
class="cmti-10">and </span><!--l. 274--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msup><mrow 
><mi 
>B</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
><mo 
class="MathClass-rel">&#x2223;</mo><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math> <span 
class="cmti-10">is the principal submatrix</span>
<span 
class="cmti-10">obtained from </span><!--l. 275--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msup><mrow 
><mi 
>B</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
></math>
<span 
class="cmti-10">by deleting its </span><!--l. 276--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>i</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>i</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></math>
<span 
class="cmti-10">rows and columns.</span>
   </div>
   <div class="newtheorem">
<!--l. 279--><p class="noindent"><span class="head">
<a 
  name="x1-3010r1"></a>
<span 
class="cmti-10">Remark 3.1.</span> </span>Let                                                                                  <!--l. 280--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>M</mi></math>
be                                                 an                                                 <!--l. 280--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>n</mi> <mo 
class="MathClass-bin">&#x00D7;</mo> <mi 
>n</mi></math>
matrix.                         The                         convention                         <!--l. 281--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>M</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>n</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>n</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mn>1</mn></math>
has been used in (<a 
href="#x1-3009r9">9<!--tex4ht:ref: detB--></a>) and hereafter.
</p>
   </div>
<!--l. 285--><p class="indent">   Before proceeding with our discussion, we pause to note that Theorem&#x00A0;<a 
href="#x1-3008r3">3.3<!--tex4ht:ref: thm-main--></a> yields
immediately a fundamental formula which can be used to compute the coefficients of
a characteristic polynomial <span class="cite">[<a 
href="#Xmami:matrixth">9</a>]</span>: </p><div class="newtheorem">
<!--l. 289--><p class="noindent"><span class="head">
<a 
  name="x1-3011r4"></a>
<span 
class="cmbx-10">Corollary 3.4.</span> </span> <span 
class="cmti-10">Write </span><!--l. 290--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mo 
>det</mo><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>B</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>x</mi><mi 
>I</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msubsup><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
  </mrow><mrow 
><mi 
>l</mi><mo 
class="MathClass-rel">=</mo><mn>0</mn></mrow><mrow 
><mi 
>n</mi></mrow></msubsup 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mi 
>l</mi></mrow></msup 
><msub><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>l</mi></mrow></msup 
></math><span 
class="cmti-10">.</span>
<span 
class="cmti-10">Then</span> </p><table class="equation"><tr><td>

<!--l. 294--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">      
<msub><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
><mo 
class="MathClass-rel">&#x2286;</mo><mi 
>n</mi></mrow></msub 
><mo 
> det</mo><mi 
>B</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
><mo 
class="MathClass-rel">&#x2223;</mo><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-3012r10"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(10)</td></tr></table>
   </div>
<!--l. 296--><p class="indent">   Let </p><table class="equation"><tr><td>

<!--l. 303--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<mi 
>K</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo>  <mfenced separators="" 
open="("  close=")" ><mtable  
equalrows="false" equalcolumns="false" class="array"><mtr><mtd 
class="array"  columnalign="center"> <msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mi 
>t</mi>  </mtd> <mtd 
class="array"  columnalign="center"><mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mi 
>a</mi></mrow><mrow 
><mn>1</mn><mn>2</mn></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mtd> <mtd 
class="array"  columnalign="center"><mo 
class="MathClass-op">&#x2026;</mo> </mtd><mtd 
class="array"  columnalign="center"><mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mi 
>a</mi></mrow><mrow 
><mn>1</mn><mi 
>n</mi></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mtd>
</mtr><mtr><mtd 
class="array"  columnalign="center"><mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mi 
>a</mi></mrow><mrow 
><mn>2</mn><mn>1</mn></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
></mtd> <mtd 
class="array"  columnalign="center">   <msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
><mi 
>t</mi>  </mtd> <mtd 
class="array"  columnalign="center"><mo 
class="MathClass-op">&#x2026;</mo></mtd><mtd 
class="array"  columnalign="center"><mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mi 
>a</mi></mrow><mrow 
><mn>2</mn><mi 
>n</mi></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mtd>
</mtr><mtr><mtd 
class="array" columnspan="4" columnalign="center"><mrow 
class="multicolumn-center"> <mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo> <mo 
class="MathClass-punc">.</mo> <mo 
class="MathClass-punc">.</mo> <mo 
class="MathClass-punc">.</mo> <mo 
class="MathClass-punc">.</mo> <mo 
class="MathClass-punc">.</mo> <mo 
class="MathClass-punc">.</mo> <mo 
class="MathClass-punc">.</mo> <mo 
class="MathClass-punc">.</mo> <mo 
class="MathClass-punc">.</mo> <mo 
class="MathClass-punc">.</mo> <mo 
class="MathClass-punc">.</mo> <mo 
class="MathClass-punc">.</mo> <mo 
class="MathClass-punc">.</mo></mrow>
</mtd></mtr><mtr><mtd 
class="array"  columnalign="center"><mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>n</mi><mn>1</mn></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
></mtd><mtd 
class="array"  columnalign="center"><mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>n</mi><mn>2</mn></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mtd><mtd 
class="array"  columnalign="center"><mo 
class="MathClass-op">&#x2026;</mo> </mtd><mtd 
class="array"  columnalign="center">   <msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
><mi 
>t</mi>  </mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable>                                                              </mfenced><mo 
class="MathClass-punc">,</mo><mstyle 
    class="label" id="x1-3013r11"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(11)</td></tr></table>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{pmatrix}&#x00A0;D&#x005F;1t&#x0026;-a&#x005F;{12}t&#x005F;2&#x0026;&#x005C;dots&#x0026;-a&#x005F;{1n}t&#x005F;n&#x005C;&#x005C;
&#x00A0;<br />-a&#x005F;{21}t&#x005F;1&#x0026;D&#x005F;2t&#x0026;&#x005C;dots&#x0026;-a&#x005F;{2n}t&#x005F;n&#x005C;&#x005C;
&#x00A0;<br />&#x005C;hdotsfor[2]{4}&#x005C;&#x005C;
&#x00A0;<br />-a&#x005F;{n1}t&#x005F;1&#x0026;-a&#x005F;{n2}t&#x005F;2&#x0026;&#x005C;dots&#x0026;D&#x005F;nt&#x005C;end{pmatrix}
</div>
</td></tr></table>
<!--l. 310--><p class="indent">   where </p><table class="equation"><tr><td>
<!--l. 313--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msub><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>i</mi><mi 
>j</mi></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mi 
>i</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><mi 
>n</mi><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-3014r12"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(12)</td></tr></table>
   Set <table class="equation"><tr><td>

<!--l. 319--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<mi 
>D</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo>  <mfrac><mrow 
><mi 
>&#x03B4;</mi></mrow> 
<mrow 
><mi 
>&#x03B4;</mi><mi 
>t</mi></mrow></mfrac><msub><mrow 
> <mfenced separators="" 
open=""  close="|" ><mo 
>det</mo><mi 
>K</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo>&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced></mrow><mrow 
><mi 
>t</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">.</mo>
</math>
</td></tr></table>
Then <table class="equation"><tr><td>
<!--l. 324--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<mi 
>D</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mo 
> det</mo><mi 
>K</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo>&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
><mo 
class="MathClass-punc">;</mo><mi 
>i</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>i</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo><mstyle 
    class="label" id="x1-3015r13"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(13)</td></tr></table>

where <!--l. 325--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>K</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
><mo 
class="MathClass-punc">;</mo><mi 
>i</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>i</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math> is the <!--l. 325--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>i</mi></math>th principal
submatrix of <!--l. 326--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>K</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>.
   Theorem &#x00A0;<a 
href="#x1-3008r3">3.3<!--tex4ht:ref: thm-main--></a> leads to <table class="equation"><tr><td>
<!--l. 334--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<mo 
>det</mo><mi 
>K</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo>&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>I</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>I</mi></mfenced></mrow></msup 
><msup><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mfenced separators="" 
open="|"  close="|" ><mi 
>I</mi></mfenced></mrow></msup 
><msub><mrow 
> <mo 
class="MathClass-op">&#x220F;</mo>
     </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>I</mi></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><msub><mrow 
> <mo 
class="MathClass-op">&#x220F;</mo>
   </mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>I</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <msub><mrow 
><mi 
>&#x03BB;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
>det</mo><msup><mrow 
><mi 
>A</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover><mo 
class="MathClass-rel">&#x2223;</mo><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-3016r14"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(14)</td></tr></table>
Note that <table class="equation"><tr><td>

<!--l. 340--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<mo 
>det</mo><mi 
>K</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo>&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>I</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>I</mi></mfenced></mrow></msup 
><msub><mrow 
> <mo 
class="MathClass-op">&#x220F;</mo>
   </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>I</mi></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><msub><mrow 
> <mo 
class="MathClass-op">&#x220F;</mo>
   </mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>I</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <msub><mrow 
><mi 
>&#x03BB;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
>det</mo><msup><mrow 
><mi 
>A</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover><mo 
class="MathClass-rel">&#x2223;</mo><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mn>0</mn><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-3017r15"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(15)</td></tr></table>
   Let <!--l. 342--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mi 
>i</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><mi 
>n</mi></math>.
Lemma &#x00A0;<a 
href="#x1-3001r1">3.1<!--tex4ht:ref: lem-per--></a> yields
<!--l. 343--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     
<mtable 
class="multline">
<mtr><mtd 
class="multline"></mtd><mtd><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msub><mrow 
><mi 
>a</mi></mrow><mrow 
><msub><mrow 
><mi 
>l</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow></msub 
><msub><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
>det</mo><mi 
>K</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>x</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo>&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
><mo 
class="MathClass-punc">;</mo><mi 
>l</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>l</mi></mrow><mo 
class="MathClass-close">)</mo></mrow>
</mtd></mtr><mtr><mtd 
class="multline"></mtd><mtd> <mo 
class="MathClass-rel">=</mo><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mo 
class="MathClass-op">&#x220F;</mo>
   </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>I</mi><mo 
class="MathClass-rel">&#x2286;</mo><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>l</mi></mrow><mo 
class="MathClass-close">}</mo></mrow></mrow></msub 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>I</mi></mfenced></mrow></msup 
><mo 
class="MathClass-op"> per</mo><!--nolimits--><msup><mrow 
><mi 
>A</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>I</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>I</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
>det</mo><msup><mrow 
><mi 
>A</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover> <mo 
class="MathClass-bin">&#x222A;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>l</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mo 
class="MathClass-rel">&#x2223;</mo><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover> <mo 
class="MathClass-bin">&#x222A;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>l</mi></mrow><mo 
class="MathClass-close">}</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-3018r16"  ></mstyle><!--endlabel--></mtd><mtd><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>1</mn><mn>6</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>   </mtd></mtr></mtable>
</math>
<!--l. 352--><p class="nopar">

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{multline}
&#x00A0;<br />&#x005C;biggl(&#x005C;sum&#x005F;{&#x005C;,i&#x005C;in&#x005C;mathbf{n}}a&#x005F;{l&#x00A0;&#x005F;i}x&#x005F;i&#x005C;biggr)
&#x00A0;<br />&#x005C;det&#x005C;mathbf{K}(t=1,x&#x005F;1,&#x005C;dots,x&#x005F;n;l&#x00A0;|l&#x00A0;)&#x005C;&#x005C;
&#x00A0;<br />=&#x005C;biggl(&#x005C;prod&#x005F;{&#x005C;,i&#x005C;in&#x005C;mathbf{n}}&#x005C;hat&#x00A0;x&#x005F;i&#x005C;biggr)
&#x00A0;<br />&#x005C;sum&#x005F;{I&#x005C;subseteq&#x005C;mathbf{n}-&#x005C;{l&#x00A0;&#x005C;}}
&#x00A0;<br />(-1)&#x02C6;{&#x005C;envert{I}}&#x005C;per&#x005C;mathbf{A}&#x02C6;{(&#x005C;lambda)}(I|I)
&#x00A0;<br />&#x005C;det&#x005C;mathbf{A}&#x02C6;{(&#x005C;lambda)}
&#x00A0;<br />(&#x005C;overline&#x00A0;I&#x005C;cup&#x005C;{l&#x00A0;&#x005C;}|&#x005C;overline&#x00A0;I&#x005C;cup&#x005C;{l&#x00A0;&#x005C;}).
&#x00A0;<br />&#x005C;label{sum-ali}
&#x00A0;<br />&#x005C;end{multline}
</div>
</td></tr></table>
<!--l. 366--><p class="indent">   By (<a 
href="#x1-2003r3">3<!--tex4ht:ref: H-cycles--></a>), (<a 
href="#x1-3005r6">6<!--tex4ht:ref: detprod--></a>), and (<a 
href="#x1-3006r7">7<!--tex4ht:ref: sum-bij--></a>), we have </p><div class="newtheorem">
<!--l. 367--><p class="noindent"><span class="head">
<a 
  name="x1-3019r5"></a>
<span 
class="cmbx-10">Proposition 3.5.</span> </span> </p><table class="equation"><tr><td>
<!--l. 371--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">      
<msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo>  <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><mn>2</mn><mi 
>n</mi></mrow></mfrac><msubsup><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
     </mrow><mrow 
><mi 
>l</mi><mo 
class="MathClass-rel">=</mo><mn>0</mn></mrow><mrow 
><mi 
>n</mi></mrow></msubsup 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mi 
>l</mi></mrow></msup 
><msub><mrow 
><mi 
>D</mi></mrow><mrow 
>
<mi 
>l</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mstyle 
    class="label" id="x1-3020r17"  ></mstyle><!--endlabel-->
</math>

</td><td class="eq-no">(17)</td></tr></table>
<span 
class="cmti-10">where </span><table class="equation"><tr><td>
<!--l. 378--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">

<msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
><mo 
class="MathClass-rel">&#x2286;</mo><mi 
>n</mi></mrow></msub 
><mi 
>D</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mn>2</mn><msub><mrow 
><mo 
class="MathClass-rel">&#x2223;</mo></mrow><mrow 
><msub><mrow 
>
<mi 
>t</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mo 
class="MathClass-rel">=</mo><mfenced separators="" 
open="{"  close="" ><mtable><mtr 
class="smallmatrix"> 
<mtd><mn>0</mn><mo 
class="MathClass-punc">,</mo></mtd> 
<mtd><!--mstyle 
class="text"--><mtext >if&#x000A0;</mtext><!--/mstyle--><mi 
>i</mi><mo 
class="MathClass-rel">&#x2208;</mo><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></mtd>
</mtr><mtr 
class="smallmatrix"> 
<mtd><mn>1</mn><mo 
class="MathClass-punc">,</mo></mtd> 
<mtd><!--mstyle 
class="text"--><mtext >otherwise</mtext><!--/mstyle--></mtd>
    </mtr>                                                                                            </mtable></mfenced><mspace width="2.83864pt" class="tmspace"/><mo 
class="MathClass-punc">,</mo><mspace width="2.83864pt" class="tmspace"/><mspace width="2.83864pt" class="tmspace"/><mi 
>i</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><mi 
>n</mi></mrow></msub 
><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-3021r18"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(18)</td></tr></table>
   </div>
   <h3 class="sectionHead"><span class="titlemark">4</span> <a 
  name="x1-40004"></a>Application</h3>
<!--l. 384--><p class="noindent">We consider here the applications of Theorems&#x00A0;<a 
href="#x1-5002r1">5.1<!--tex4ht:ref: th-info-ow-ow--></a> and &#x00A0;<a 
href="#x1-5003r2">5.2<!--tex4ht:ref: th-weak-ske-owf--></a> to a complete multipartite graph <!--l. 386--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>K</mi></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-op">&#x2026;</mo><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>p</mi></mrow></msub 
></mrow></msub 
></math>.
It can be shown that the number of spanning trees of <!--l. 387--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>K</mi></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-op">&#x2026;</mo><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>p</mi></mrow></msub 
></mrow></msub 
></math> may
be written </p><table class="equation"><tr><td>

<!--l. 392--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<mi 
>T</mi> <mo 
class="MathClass-rel">=</mo> <msup><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>p</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>2</mn></mrow></msup 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x220F;</mo>
    </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>p</mi></mrow></msubsup 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>n</mi></mrow><mrow 
>
<mi 
>i</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msup 
><mstyle 
    class="label" id="x1-4001r19"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(19)</td></tr></table>
where <table class="equation"><tr><td>

<!--l. 396--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<mi 
>n</mi> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <mo 
class="MathClass-rel">&#x22EF;</mo> <mo 
class="MathClass-bin">+</mo> <msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>p</mi></mrow></msub 
><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-4002r20"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(20)</td></tr></table>
   It follows from Theorems&#x00A0;<a 
href="#x1-5002r1">5.1<!--tex4ht:ref: th-info-ow-ow--></a> and &#x00A0;<a 
href="#x1-5003r2">5.2<!--tex4ht:ref: th-weak-ske-owf--></a> that <table class="equation"><tr><td>

<!--l. 408--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow> <msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo>  <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><mn>2</mn><mi 
>n</mi></mrow></mfrac><msubsup><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
     </mrow><mrow 
><mi 
>l</mi><mo 
class="MathClass-rel">=</mo><mn>0</mn></mrow><mrow 
><mi 
>n</mi></mrow></msubsup 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mi 
>l</mi></mrow></msup 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>l</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mi 
>p</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>2</mn></mrow></msup 
><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
     </mrow><mrow 
><msub><mrow 
><mi 
>l</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-bin">+</mo><mo 
class="MathClass-rel">&#x22EF;</mo><mo 
class="MathClass-bin">+</mo><msub><mrow 
><mi 
>l</mi></mrow><mrow 
><mi 
>p</mi></mrow></msub 
><mo 
class="MathClass-rel">=</mo><mi 
>l</mi></mrow></msub 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x220F;</mo>
            </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>p</mi></mrow></msubsup 
><mfenced separators="" 
open="(" close=")"><mfrac linethickness="0.0pt"><mrow><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow>
<mrow><msub><mrow 
><mi 
>l</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow></mfrac></mfenced>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>     <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-punc">&#x22C5;</mo> <msup><mrow 
><mrow><mo 
class="MathClass-open">[</mo><mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>l</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>l</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">]</mo></mrow></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mi 
>l</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
>
       </mrow></msup 
> <mo 
class="MathClass-punc">&#x22C5;</mo><mrow><mo 
class="MathClass-open">[</mo><mrow><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>l</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
> <mo 
class="MathClass-bin">&#x2212;</mo><msubsup><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
  </mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>p</mi></mrow></msubsup 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>n</mi></mrow><mrow 
>
<mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>l</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
></mrow><mo 
class="MathClass-close">]</mo></mrow><mo 
class="MathClass-punc">.</mo>
    </mtd></mtr></mtable>                                                                                    <mstyle 
    class="label" id="x1-4003r21"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(21)</td></tr></table>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
...&#x00A0;&#x005C;binom{n&#x005F;i}{l&#x00A0;&#x005F;i}&#x005C;&#x005C;
</div>
</td></tr></table>
<!--l. 412--><p class="indent">   and </p><table class="equation"><tr><td>
<!--l. 423--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow> <msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><mn>2</mn></mrow></mfrac><msubsup><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
    </mrow><mrow 
><mi 
>l</mi><mo 
class="MathClass-rel">=</mo><mn>0</mn></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msubsup 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mi 
>l</mi></mrow></msup 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>l</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mi 
>p</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>2</mn></mrow></msup 
><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
     </mrow><mrow 
><msub><mrow 
><mi 
>l</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-bin">+</mo><mo 
class="MathClass-rel">&#x22EF;</mo><mo 
class="MathClass-bin">+</mo><msub><mrow 
><mi 
>l</mi></mrow><mrow 
><mi 
>p</mi></mrow></msub 
><mo 
class="MathClass-rel">=</mo><mi 
>l</mi></mrow></msub 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x220F;</mo>
            </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>p</mi></mrow></msubsup 
><mfenced separators="" 
open="(" close=")"><mfrac linethickness="0.0pt"><mrow><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow>
<mrow><msub><mrow 
><mi 
>l</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow></mfrac></mfenced>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>     <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-punc">&#x22C5;</mo> <msup><mrow 
><mrow><mo 
class="MathClass-open">[</mo><mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>l</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>l</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">]</mo></mrow></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mi 
>l</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
>
       </mrow></msup 
> <mfenced separators="" 
open="("  close=")" ><mn>1</mn> <mo 
class="MathClass-bin">&#x2212;</mo> <mfrac><mrow 
><msub><mrow 
><mi 
>l</mi></mrow><mrow 
><mi 
>p</mi></mrow></msub 
></mrow> 
<mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>p</mi></mrow></msub 
></mrow></mfrac></mfenced> <mrow><mo 
class="MathClass-open">[</mo><mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>l</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>p</mi></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>l</mi></mrow><mrow 
><mi 
>p</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">]</mo></mrow><mo 
class="MathClass-punc">.</mo>
    </mtd></mtr></mtable>                                                                                    <mstyle 
    class="label" id="x1-4004r22"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(22)</td></tr></table>
   The enumeration of <!--l. 425--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
></math>
in a <!--l. 425--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>K</mi></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-rel">&#x22EF;</mo><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mi 
>p</mi></mrow></msub 
></mrow></msub 
></math>
graph can also be carried out by Theorem &#x00A0;<a 
href="#x1-9005r2">7.2<!--tex4ht:ref: thm-H-param--></a> or &#x00A0;<a 
href="#x1-9012r3">7.3<!--tex4ht:ref: thm-asym--></a> together with the algebraic
method of (<a 
href="#x1-2002r2">2<!--tex4ht:ref: multdef--></a>). Some elegant representations may be obtained. For example, <!--l. 428--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
></math> in a <!--l. 429--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>K</mi></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
></mrow></msub 
></math> graph
may be written <table class="equation"><tr><td>

<!--l. 439--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow> <msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo><mrow 
class="split-mtd"></mrow>  <mfrac><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mi 
>!</mi><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
><mi 
>!</mi><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
><mi 
>!</mi></mrow>
<mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
></mrow></mfrac><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
           </mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mfenced separators="" 
open="["  close="" ><mfenced separators="" 
open="(" close=")"><mfrac linethickness="0.0pt"><mrow><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
></mrow>
 <mrow><mi 
>i</mi></mrow></mfrac></mfenced><mfenced separators="" 
open="(" close=")"><mfrac linethickness="0.0pt"><mrow>      <msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mrow> 
<mrow><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <mi 
>i</mi></mrow></mfrac></mfenced><mfenced separators="" 
open="(" close=")"><mfrac linethickness="0.0pt"><mrow>     <msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
></mrow> 
<mrow><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <mi 
>i</mi></mrow></mfrac></mfenced></mfenced>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>       <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-bin">+</mo> <mfenced separators="" 
open=""  close="]" ><mfenced separators="" 
open="(" close=")"><mfrac linethickness="0.0pt"><mrow><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>1</mn></mrow>
    <mrow><mi 
>i</mi></mrow></mfrac></mfenced><mfenced separators="" 
open="(" close=")"><mfrac linethickness="0.0pt"><mrow>      <msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>1</mn></mrow> 
<mrow><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <mi 
>i</mi></mrow></mfrac></mfenced><mfenced separators="" 
open="(" close=")"><mfrac linethickness="0.0pt"><mrow>   <msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>1</mn></mrow> 
<mrow><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <mi 
>i</mi></mrow></mfrac></mfenced></mfenced><mo 
class="MathClass-punc">.</mo>
    </mtd></mtr></mtable>                                                                                    <mstyle 
    class="label" id="x1-4005r23"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(23)</td></tr></table>
   <h3 class="sectionHead"><span class="titlemark">5</span> <a 
  name="x1-50005"></a>Secret Key Exchanges</h3>
<!--l. 444--><p class="noindent">Modern cryptography is fundamentally concerned with the problem of secure private
communication. A Secret Key Exchange is a protocol where Alice and Bob, having no
secret information in common to start, are able to agree on a common secret key,
conversing over a public channel. The notion of a Secret Key Exchange protocol was
first introduced in the seminal paper of Diffie and Hellman <span class="cite">[<a 
href="#Xdihe:newdir">1</a>]</span>. <span class="cite">[<a 
href="#Xdihe:newdir">1</a>]</span> presented a
concrete implementation of a Secret Key Exchange protocol, dependent on a specific
assumption (a variant on the discrete log), specially tailored to yield Secret Key
Exchange. Secret Key Exchange is of course trivial if trapdoor permutations
exist. However, there is no known implementation based on a weaker general
assumption.
</p><!--l. 457--><p class="indent">   The concept of an informationally one-way function was introduced in <span class="cite">[<a 
href="#Ximlelu:oneway">5</a>]</span>. We
give only an informal definition here:
</p>

   <div class="newtheorem">
<!--l. 460--><p class="noindent"><span class="head">
<a 
  name="x1-5001r1"></a>
<span 
class="cmbx-10">Definition 5.1.</span> </span>A polynomial time computable function <!--l. 461--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>f</mi> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">{</mo><mrow><msub><mrow 
><mi 
>f</mi></mrow><mrow 
><mi 
>k</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">}</mo></mrow></math>
is informationally one-way if there is no probabilistic polynomial time algorithm
which (with probability of the form <!--l. 463--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mn>1</mn> <mo 
class="MathClass-bin">&#x2212;</mo> <msup><mrow 
><mi 
>k</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mi 
>e</mi></mrow></msup 
></math>
for some <!--l. 463--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>e</mi> <mo 
class="MathClass-rel">&#x003E;</mo> <mn>0</mn></math>)
returns on input <!--l. 464--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>y</mi> <mo 
class="MathClass-rel">&#x2208;</mo><msup><mrow 
><mrow><mo 
class="MathClass-open">{</mo><mrow><mn>0</mn><mo 
class="MathClass-punc">,</mo><mn>1</mn></mrow><mo 
class="MathClass-close">}</mo></mrow></mrow><mrow 
><mi 
>k</mi></mrow></msup 
></math>
a random element of <!--l. 464--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msup><mrow 
><mi 
>f</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>.
</p>
   </div>
<!--l. 466--><p class="indent">   In the non-uniform setting <span class="cite">[<a 
href="#Ximlelu:oneway">5</a>]</span> show that these are not weaker than one-way
functions: </p><div class="newtheorem">
<!--l. 468--><p class="noindent"><span class="head">
<a 
  name="x1-5002r1"></a>
<span 
class="cmbx-10">Theorem 5.1 (</span><span class="cite"><span 
class="cmbx-10">[</span><a 
href="#Ximlelu:oneway"><span 
class="cmbx-10">5</span></a><span 
class="cmbx-10">]</span></span> <span 
class="cmbx-10">(non-uniform)).</span> </span>   <span 
class="cmti-10">The   existence   of   informationally</span>
<span 
class="cmti-10">one-way functions implies the existence of one-way functions.</span>
</p>
   </div>
<!--l. 473--><p class="indent">   We will stick to the convention introduced above of saying &#x201C;non-uniform&#x201D; before
the theorem statement when the theorem makes use of non-uniformity. It should be
understood that if nothing is said then the result holds for both the uniform and the
non-uniform models.
</p><!--l. 479--><p class="indent">   It now follows from Theorem&#x00A0;<a 
href="#x1-5002r1">5.1<!--tex4ht:ref: th-info-ow-ow--></a> that
</p>
   <div class="newtheorem">
<!--l. 481--><p class="noindent"><span class="head">
<a 
  name="x1-5003r2"></a>
<span 
class="cmbx-10">Theorem 5.2 (non-uniform).</span> </span> <span 
class="cmti-10">Weak SKE implies the existence of a one-way</span>
<span 
class="cmti-10">function.</span>
</p>
   </div>
<!--l. 485--><p class="indent">   More recently, the polynomial-time, interior point algorithms for linear
programming have been extended to the case of convex quadratic programs
<span class="cite">[<a 
href="#Xmoad:quadpro">11</a>,&#x00A0;<a 
href="#Xye:intalg">13</a>]</span>, certain linear complementarity problems <span class="cite">[<a 
href="#Xkomiyo:lincomp">7</a>,&#x00A0;<a 
href="#Xmiyoki:lincomp">10</a>]</span>, and the nonlinear
complementarity problem <span class="cite">[<a 
href="#Xkomiyo:unipfunc">6</a>]</span>. The connection between these algorithms and
the classical Newton method for nonlinear equations is well explained in
<span class="cite">[<a 
href="#Xkomiyo:lincomp">7</a>]</span>.
</p>

   <h3 class="sectionHead"><span class="titlemark">6</span> <a 
  name="x1-60006"></a>Review</h3>
<!--l. 496--><p class="noindent">We begin our discussion with the following definition:
</p>
   <div class="newtheorem">
<!--l. 498--><p class="noindent"><span class="head">
<a 
  name="x1-6001r1"></a>
<span 
class="cmbx-10">Definition 6.1.</span> </span>
</p><!--l. 500--><p class="indent">   A function <!--l. 500--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>H</mi> <mo 
class="MathClass-punc">:</mo> <msup><mrow 
><mi 
>&#x211C;</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
> <mo 
class="MathClass-rel">&#x2192;</mo><msup><mrow 
><mi 
>&#x211C;</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
></math>
is said to be <span 
class="cmti-10">B-differentiable </span>at the point <!--l. 501--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>z</mi></math>
if (i)&#x00A0;<!--l. 501--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>H</mi></math>
is Lipschitz continuous in a neighborhood of <!--l. 502--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>z</mi></math>,
and (ii)&#x00A0; there exists a positive homogeneous function <!--l. 503--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>B</mi><mi 
>H</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">:</mo> <msup><mrow 
><mi 
>&#x211C;</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
> <mo 
class="MathClass-rel">&#x2192;</mo><msup><mrow 
><mi 
>&#x211C;</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
></math>,
called the <span 
class="cmti-10">B-derivative </span>of <!--l. 504--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>H</mi></math>
at <!--l. 504--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>z</mi></math>,
such that <!--l. 505--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">      <mrow 
>
                         <msub><mrow 
><mo 
>lim</mo></mrow><mrow 
><mi 
>v</mi><mo 
class="MathClass-rel">&#x2192;</mo><mn>0</mn></mrow></msub 
><mfrac><mrow 
><mi 
>H</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>v</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>H</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>B</mi><mi 
>H</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>v</mi></mrow> 
              <mrow 
> <mfenced separators="" 
open="|"  close="|" ><mi 
>v</mi></mfenced></mrow></mfrac>               <mo 
class="MathClass-rel">=</mo> <mn>0</mn><mo 
class="MathClass-punc">.</mo>
</mrow></math>
The function <!--l. 506--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>H</mi></math>
is <span 
class="cmti-10">B-differentiable in set </span><!--l. 506--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>S</mi></math>
if it is B-differentiable at every point in <!--l. 507--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>S</mi></math>.
The B-derivative <!--l. 507--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>B</mi><mi 
>H</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>

is said to be <span 
class="cmti-10">strong </span>if <!--l. 509--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">      <mrow 
>
              <msub><mrow 
><mo 
>lim</mo></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>v</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>v</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-rel">&#x2192;</mo><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>0</mn><mo 
class="MathClass-punc">,</mo><mn>0</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><mfrac><mrow 
><mi 
>H</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>v</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>H</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi> <mo 
class="MathClass-bin">+</mo> <msup><mrow 
><mi 
>v</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>B</mi><mi 
>H</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>v</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msup><mrow 
><mi 
>v</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
                     <mrow 
> <mfenced separators="" 
open="|"  close="|" ><mi 
>v</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msup><mrow 
><mi 
>v</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mfenced></mrow></mfrac>            <mo 
class="MathClass-rel">=</mo> <mn>0</mn><mo 
class="MathClass-punc">.</mo>
</mrow></math>
</p>
   </div>
   <div class="newtheorem">
<!--l. 514--><p class="noindent"><span class="head">
<a 
  name="x1-6002r1"></a>
<span 
class="cmbx-10">Lemma 6.1.</span> </span> <span 
class="cmti-10">There exists a smooth function </span><!--l. 514--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math> <span 
class="cmti-10">defined for</span>
<!--l. 515--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi></mfenced> <mo 
class="MathClass-rel">&#x003E;</mo> <mn>1</mn> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
>a</mi></math>
<span 
class="cmti-10">satisfying the following properties</span>:
     </p><ol type="1" class="enumerate1" start="1" 
>
     <li class="enumerate"><a 
  name="x1-6004x1"></a><!--l. 518--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
     <span 
class="cmti-10">is bounded above and below by positive constants </span><!--l. 519--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>c</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
> <mo 
class="MathClass-rel">&#x2264;</mo> <msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2264;</mo> <msub><mrow 
><mi 
>c</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></math><span 
class="cmti-10">.</span>
     </li>
     <li class="enumerate"><a 
  name="x1-6006x2"></a><span 
class="cmti-10">If </span><!--l. 520--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi></mfenced> <mo 
class="MathClass-rel">&#x003E;</mo> <mn>1</mn></math><span 
class="cmti-10">,</span>
     <span 
class="cmti-10">then </span><!--l. 520--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mn>1</mn></math><span 
class="cmti-10">.</span>
     </li>
     <li class="enumerate"><a 
  name="x1-6008x3"></a><span 
class="cmti-10">For all </span><!--l. 521--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>z</mi></math>
     <span 
class="cmti-10">in the domain of </span><!--l. 521--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
></math><span 
class="cmti-10">,</span>
     <!--l. 521--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x0394;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mo 
> ln</mo><!--nolimits--><msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
> <mo 
class="MathClass-rel">&#x2265;</mo> <mn>0</mn></math><span 
class="cmti-10">.</span>
     </li>
     <li class="enumerate"><a 
  name="x1-6010x4"></a><span 
class="cmti-10">If </span><!--l. 522--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mn>1</mn> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
>a</mi> <mo 
class="MathClass-rel">&#x003C;</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi></mfenced> <mo 
class="MathClass-rel">&#x003C;</mo> <mn>1</mn> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>a</mi></math><span 
class="cmti-10">,</span>
     <span 
class="cmti-10">then </span><!--l. 522--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x0394;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mo 
> ln</mo><!--nolimits--><msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
> <mo 
class="MathClass-rel">&#x2265;</mo> <msub><mrow 
><mi 
>c</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
> <mo 
class="MathClass-rel">&#x003E;</mo> <mn>0</mn></math><span 
class="cmti-10">.</span></li></ol>
   </div>
<!--l. 527--><p class="indent">
</p>
   <div class="proof">
<!--l. 528--><p class="indent">   <span class="head">

<span 
class="cmti-10">Proof.</span> </span>We choose <!--l. 528--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
to be a radial function depending only on <!--l. 528--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>r</mi> <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi></mfenced></math>.
Let <!--l. 529--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>h</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2265;</mo> <mn>0</mn></math>
be a suitable smooth function satisfying <!--l. 529--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>h</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2265;</mo> <msub><mrow 
><mi 
>c</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
></math>
for <!--l. 530--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mn>1</mn> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
>a</mi> <mo 
class="MathClass-rel">&#x003C;</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi></mfenced> <mo 
class="MathClass-rel">&#x003C;</mo> <mn>1</mn> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>a</mi></math>,
and <!--l. 530--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>h</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mn>0</mn></math>
for <!--l. 530--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi></mfenced> <mo 
class="MathClass-rel">&#x003E;</mo> <mn>1</mn> <mo 
class="MathClass-bin">&#x2212;</mo><mfrac><mrow> <mi 
>a</mi></mrow> 
<mrow><mn>2</mn></mrow></mfrac> </math>.
The radial Laplacian <!--l. 532--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">      <mrow 
>
                             <msub><mrow 
><mi 
>&#x0394;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mo 
> ln</mo><!--nolimits--><msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="("  close=")" > <mfrac><mrow 
><msup><mrow 
><mi 
>d</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
></mrow>
<mrow 
><mi 
>d</mi><msup><mrow 
><mi 
>r</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
></mrow></mfrac> <mo 
class="MathClass-bin">+</mo> <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><mi 
>r</mi></mrow></mfrac> <mfrac><mrow 
><mi 
>d</mi></mrow> 
<mrow 
><mi 
>d</mi><mi 
>r</mi></mrow></mfrac></mfenced><mo 
>ln</mo><!--nolimits--><msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow>
</mrow></math>
has smooth coefficients for <!--l. 534--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>r</mi> <mo 
class="MathClass-rel">&#x003E;</mo> <mn>1</mn> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
>a</mi></math>.
Therefore,  we  may  apply  the  existence  and  uniqueness  theory  for  ordinary
differential equations. Simply let <!--l. 536--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mo 
>ln</mo><!--nolimits--><msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
be the solution of the differential equation <!--l. 538--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">      <mrow 
>
                                  <mfenced separators="" 
open="("  close=")" > <mfrac><mrow 
><msup><mrow 
><mi 
>d</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
></mrow>
<mrow 
><mi 
>d</mi><msup><mrow 
><mi 
>r</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
></mrow></mfrac> <mo 
class="MathClass-bin">+</mo> <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><mi 
>r</mi></mrow></mfrac> <mfrac><mrow 
><mi 
>d</mi></mrow> 
<mrow 
><mi 
>d</mi><mi 
>r</mi></mrow></mfrac></mfenced><mo 
>ln</mo><!--nolimits--><msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>h</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow>
</mrow></math>
with initial conditions given by <!--l. 539--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mo 
>ln</mo><!--nolimits--><msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mn>0</mn></math>
and <!--l. 540--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mo 
>ln</mo><!--nolimits--><msubsup><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msubsup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mn>0</mn></math>.
</p><!--l. 542--><p class="indent">   Next, let <!--l. 542--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></math>
be a finite collection of pairwise disjoint disks, all of which
are contained in the unit disk centered at the origin in <!--l. 544--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>C</mi></math>. We assume that <!--l. 544--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>z</mi> <mo 
class="MathClass-rel">&#x2223;</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>z</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></mfenced> <mo 
class="MathClass-rel">&#x003C;</mo> <mi 
>&#x03B4;</mi></mrow><mo 
class="MathClass-close">}</mo></mrow></math>. Suppose that <!--l. 545--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">

<msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>a</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>denotes the smaller concentric
disk <!--l. 545--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>a</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>z</mi> <mo 
class="MathClass-rel">&#x2223;</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>z</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></mfenced> <mo 
class="MathClass-rel">&#x2264;</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><mn>1</mn> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
>a</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>&#x03B4;</mi></mrow><mo 
class="MathClass-close">}</mo></mrow></math>. We define a
smooth weight function <!--l. 547--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
for <!--l. 547--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>z</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>C</mi> <mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mo 
class="MathClass-op">&#x22C3;</mo>
  </mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
><msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>a</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math> by
setting <!--l. 547--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mn>1</mn></math>
when <!--l. 548--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>z</mi><mo 
class="MathClass-rel">&#x2209;</mo><msub><mrow 
><mo 
class="MathClass-op">&#x22C3;</mo>
  </mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
><msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></math> and
<!--l. 548--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">     <msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>z</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-bin">/</mo><mi 
>&#x03B4;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math> when <!--l. 549--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>z</mi></math> is an element of <!--l. 549--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></math>. It follows from
Lemma&#x00A0;<a 
href="#x1-6002r1">6.1<!--tex4ht:ref: limbog--></a> that <!--l. 550--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
></math>
satisfies the properties:
     </p><ol type="1" class="enumerate1" start="1" 
>
     <li class="enumerate"><a 
  name="x1-6012x1"></a><!--l. 553--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
     is bounded above and below by positive constants <!--l. 554--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>c</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
> <mo 
class="MathClass-rel">&#x2264;</mo> <msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2264;</mo> <msub><mrow 
><mi 
>c</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></math>.
     </li>
     <li class="enumerate"><a 
  name="x1-6014x2"></a><!--l. 555--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x0394;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mo 
> ln</mo><!--nolimits--><msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
> <mo 
class="MathClass-rel">&#x2265;</mo> <mn>0</mn></math>
     for all <!--l. 556--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>z</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>C</mi> <mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mo 
class="MathClass-op">&#x22C3;</mo>
  </mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
><msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>a</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>,
     the domain where the function <!--l. 557--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
></math>
     is defined.
     </li>
     <li class="enumerate"><a 
  name="x1-6016x3"></a><!--l. 558--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x0394;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mo 
> ln</mo><!--nolimits--><msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
> <mo 
class="MathClass-rel">&#x2265;</mo> <msub><mrow 
><mi 
>c</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
><msup><mrow 
><mi 
>&#x03B4;</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mn>2</mn></mrow></msup 
></math>
     when <!--l. 559--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mrow><mo 
class="MathClass-open">(</mo><mrow><mn>1</mn> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
>a</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>&#x03B4;</mi> <mo 
class="MathClass-rel">&#x003C;</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>z</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></mfenced> <mo 
class="MathClass-rel">&#x003C;</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><mn>1</mn> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>a</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>&#x03B4;</mi></math>.</li></ol>
<!--l. 561--><p class="noindent">Let <!--l. 561--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></math> denote the
annulus <!--l. 561--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">{</mo><mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>1</mn> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
>a</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>&#x03B4;</mi> <mo 
class="MathClass-rel">&#x003C;</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>z</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></mfenced> <mo 
class="MathClass-rel">&#x003C;</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><mn>1</mn> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>a</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>&#x03B4;</mi></mrow><mo 
class="MathClass-close">}</mo></mrow></math>, and set <!--l. 562--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>A</mi> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x22C3;</mo>
  </mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
><msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></math>. The properties (<a 
href="#x1-6014x2">2<!--tex4ht:ref: d:over--></a>)
and (<a 
href="#x1-6016x3">3<!--tex4ht:ref: d:ad--></a>) of <!--l. 563--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
></math> may be
summarized as <!--l. 564--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>&#x0394;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mo 
> ln</mo><!--nolimits--><msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
> <mo 
class="MathClass-rel">&#x2265;</mo> <msub><mrow 
><mi 
>c</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
><msup><mrow 
><mi 
>&#x03B4;</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mn>2</mn></mrow></msup 
><msub><mrow 
><mi 
>&#x03C7;</mi></mrow><mrow 
><mi 
>A</mi></mrow></msub 
></math>, where <!--l. 565--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>&#x03C7;</mi></mrow><mrow 
><mi 
>A</mi></mrow></msub 
></math> is the characteristic
function of <!--l. 565--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>A</mi></math>.   &#x25A1;
</p>
   </div>
<!--l. 568--><p class="indent">   Suppose that <!--l. 568--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>&#x03B1;</mi></math>
is a nonnegative real constant. We apply Proposition&#x00A0;<a 
href="#x1-3019r5">3.5<!--tex4ht:ref: prop:eg--></a> with <!--l. 569--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>&#x03A6;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msup><mrow 
><mi 
>e</mi></mrow><mrow 
><mi 
>&#x03B1;</mi><msup><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
>
       </mrow></msup 
></math>. If <!--l. 570--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>u</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msubsup><mrow 
><mi 
>C</mi></mrow><mrow 
><mn>0</mn></mrow><mrow 
><mi 
>&#x221E;</mi></mrow></msubsup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
> <mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mo 
class="MathClass-op">&#x22C3;</mo>
  </mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
><msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>a</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></math>, assume
that <!--l. 570--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
><!--span 
class="htf-calligraphy"-->D<!--/span--></mi></math>
is a bounded domain containing the support of <!--l. 571--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>u</mi></math> and <!--l. 571--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>A</mi> <mo 
class="MathClass-rel">&#x2282;</mo><mi 
><!--span 
class="htf-calligraphy"-->D<!--/span--></mi><mo 
class="MathClass-rel">&#x2282;</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
> <mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mo 
class="MathClass-op">&#x22C3;</mo>
  </mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
><msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>a</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>. A calculation

gives <!--l. 573--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">        <mrow 
><msub><mrow 
>
        <mo 
class="MathClass-op">&#x222B;</mo>
      </mrow><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->D<!--/span--></mi></mrow></msub 
><msup><mrow 
><mfenced separators="" 
open="|"  close="|" ><mover accent="false" 
class="mml-overline"><mrow><mi 
>&#x2202;</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover><mi 
>u</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
><msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
>
<mn>0</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msup><mrow 
><mi 
>e</mi></mrow><mrow 
><mi 
>&#x03B1;</mi><msup><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
>
       </mrow></msup 
> <mo 
class="MathClass-rel">&#x2265;</mo> <msub><mrow 
><mi 
>c</mi></mrow><mrow 
><mn>4</mn></mrow></msub 
><mi 
>&#x03B1;</mi><msub><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->D<!--/span--></mi></mrow></msub 
><msup><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>u</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
><msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
>
<mn>0</mn></mrow></msub 
><msup><mrow 
><mi 
>e</mi></mrow><mrow 
><mi 
>&#x03B1;</mi><msup><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
>
       </mrow></msup 
> <mo 
class="MathClass-bin">+</mo> <msub><mrow 
><mi 
>c</mi></mrow><mrow 
><mn>5</mn></mrow></msub 
><msup><mrow 
><mi 
>&#x03B4;</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mn>2</mn></mrow></msup 
><msub><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
   </mrow><mrow 
><mi 
>A</mi></mrow></msub 
><msup><mrow 
> <mfenced separators="" 
open="|"  close="|" ><mi 
>u</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
><msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
>
<mn>0</mn></mrow></msub 
><msup><mrow 
><mi 
>e</mi></mrow><mrow 
><mi 
>&#x03B1;</mi><msup><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
>
       </mrow></msup 
><mo 
class="MathClass-punc">.</mo>
</mrow></math>
</p><!--l. 577--><p class="indent">   The boundedness, property (<a 
href="#x1-6012x1">1<!--tex4ht:ref: boundab--></a>) of <!--l. 577--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
></math>, then yields <!--l. 578--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
><msub><mrow 
>
          <mo 
class="MathClass-op">&#x222B;</mo>
            </mrow><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->D<!--/span--></mi></mrow></msub 
><msup><mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover accent="false" 
class="mml-overline"><mrow><mi 
>&#x2202;</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover><mi 
>u</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
><msup><mrow 
><mi 
>e</mi></mrow><mrow 
><mi 
>&#x03B1;</mi><msup><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
>
       </mrow></msup 
> <mo 
class="MathClass-rel">&#x2265;</mo> <msub><mrow 
><mi 
>c</mi></mrow><mrow 
><mn>6</mn></mrow></msub 
><mi 
>&#x03B1;</mi><msub><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->D<!--/span--></mi></mrow></msub 
><msup><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>u</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
><msup><mrow 
><mi 
>e</mi></mrow><mrow 
><mi 
>&#x03B1;</mi><msup><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
>
       </mrow></msup 
> <mo 
class="MathClass-bin">+</mo> <msub><mrow 
><mi 
>c</mi></mrow><mrow 
><mn>7</mn></mrow></msub 
><msup><mrow 
><mi 
>&#x03B4;</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mn>2</mn></mrow></msup 
><msub><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
   </mrow><mrow 
><mi 
>A</mi></mrow></msub 
><msup><mrow 
> <mfenced separators="" 
open="|"  close="|" ><mi 
>u</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
><msup><mrow 
><mi 
>e</mi></mrow><mrow 
><mi 
>&#x03B1;</mi><msup><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
>
       </mrow></msup 
><mo 
class="MathClass-punc">.</mo>
</mrow></math>
</p><!--l. 582--><p class="indent">   Let <!--l. 582--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>B</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math> be the
set of blocks of <!--l. 582--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>&#x039B;</mi></mrow><mrow 
><mi 
>X</mi></mrow></msub 
></math>
and let <!--l. 583--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>b</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>B</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced></math>.
If <!--l. 583--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>&#x03C6;</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>Q</mi></mrow><mrow 
><mi 
>X</mi></mrow></msub 
></math> then <!--l. 584--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>&#x03C6;</mi></math> is constant on the
blocks of <!--l. 584--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>&#x039B;</mi></mrow><mrow 
><mi 
>X</mi></mrow></msub 
></math>.
</p><table class="equation"><tr><td>

<!--l. 589--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<msub><mrow 
><mi 
>P</mi></mrow><mrow 
><mi 
>X</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>&#x03C6;</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>M</mi> <mo 
class="MathClass-rel">&#x2223;</mo> <msub><mrow 
><mi 
>&#x039B;</mi></mrow><mrow 
><mi 
>&#x03C6;</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>&#x039B;</mi></mrow><mrow 
><mi 
>X</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">}</mo></mrow><mo 
class="MathClass-punc">,</mo><mspace width="2em" class="qquad"/><msub><mrow 
><mi 
>Q</mi></mrow><mrow 
><mi 
>X</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>&#x03C6;</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>M</mi> <mo 
class="MathClass-rel">&#x2223;</mo> <msub><mrow 
><mi 
>&#x039B;</mi></mrow><mrow 
><mi 
>&#x03C6;</mi></mrow></msub 
> <mo 
class="MathClass-rel">&#x2265;</mo> <msub><mrow 
><mi 
>&#x039B;</mi></mrow><mrow 
><mi 
>X</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">}</mo></mrow><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-6017r24"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(24)</td></tr></table>
If <!--l. 590--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>&#x039B;</mi></mrow><mrow 
><mi 
>&#x03C6;</mi></mrow></msub 
> <mo 
class="MathClass-rel">&#x2265;</mo> <msub><mrow 
><mi 
>&#x039B;</mi></mrow><mrow 
><mi 
>X</mi></mrow></msub 
></math> then <!--l. 591--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>&#x039B;</mi></mrow><mrow 
><mi 
>&#x03C6;</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>&#x039B;</mi></mrow><mrow 
><mi 
>Y</mi> </mrow></msub 
></math> for some <!--l. 591--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>Y</mi> <mo 
class="MathClass-rel">&#x2265;</mo> <mi 
>X</mi></math> so that <!--l. 592--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
>
                         <msub><mrow 
><mi 
>Q</mi></mrow><mrow 
><mi 
>X</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x22C3;</mo>
  </mrow><mrow 
><mi 
>Y</mi> <mo 
class="MathClass-rel">&#x2265;</mo><mi 
>X</mi></mrow></msub 
><msub><mrow 
><mi 
>P</mi></mrow><mrow 
><mi 
>Y</mi> </mrow></msub 
><mo 
class="MathClass-punc">.</mo>
</mrow></math> Thus by M&#x00F6;bius

inversion <!--l. 594--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">       <mrow 
>
                                 <mfenced separators="" 
open="|"  close="|" ><msub><mrow 
><mi 
>P</mi></mrow><mrow 
><mi 
>Y</mi> </mrow></msub 
></mfenced> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>X</mi><mo 
class="MathClass-rel">&#x2265;</mo><mi 
>Y</mi> </mrow></msub 
><mi 
>&#x03BC;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>Y</mi><mo 
class="MathClass-punc">,</mo><mi 
>X</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mfenced separators="" 
open="|"  close="|" ><msub><mrow 
><mi 
>Q</mi></mrow><mrow 
><mi 
>X</mi></mrow></msub 
></mfenced> <mo 
class="MathClass-punc">.</mo>
</mrow></math> Thus there
is a bijection from <!--l. 595--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>Q</mi></mrow><mrow 
><mi 
>X</mi></mrow></msub 
></math>
to <!--l. 595--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msup><mrow 
><mi 
>W</mi></mrow><mrow 
><mi 
>B</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
></math>. In
particular <!--l. 596--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mfenced separators="" 
open="|"  close="|" ><msub><mrow 
><mi 
>Q</mi></mrow><mrow 
><mi 
>X</mi></mrow></msub 
></mfenced> <mo 
class="MathClass-rel">=</mo> <msup><mrow 
><mi 
>w</mi></mrow><mrow 
><mi 
>b</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
></math>.
   Next note that <!--l. 598--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>b</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><mo 
> dim</mo><!--nolimits--><mi 
>X</mi></math>. We see
this by choosing a basis for <!--l. 599--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>X</mi></math>
consisting of vectors <!--l. 599--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msup><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>k</mi></mrow></msup 
></math>
defined by <!--l. 600--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">        <mrow 
><msubsup><mrow 
>
<mi 
>v</mi></mrow><mrow 
><mi 
>i</mi></mrow><mrow 
><mi 
>k</mi></mrow></msubsup 
> <mo 
class="MathClass-rel">=</mo>  <mfenced separators="" 
open="{"  close="" ><mtable  
equalrows="false" equalcolumns="false" class="array"><mtr><mtd 
class="array"  columnalign="left"><mn>1</mn></mtd><mtd 
class="array"  columnalign="left"><!--mstyle 
class="text"--><mtext >if&#x000A0;</mtext><!--mstyle 
class="math"--><mi 
>i</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>&#x039B;</mi></mrow><mrow 
><mi 
>k</mi></mrow></msub 
><!--/mstyle--><mtext ></mtext><!--/mstyle--><mo 
class="MathClass-punc">,</mo> </mtd>
</mtr><mtr><mtd 
class="array"  columnalign="left"><mn>0</mn></mtd><mtd 
class="array"  columnalign="left"><!--mstyle 
class="text"--><mtext >otherwise.</mtext><!--/mstyle--></mtd></mtr> <!--@{}l@{\quad }l@{}--></mtable>                                                                                         </mfenced>
</mrow></math>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;[v&#x02C6;{k}&#x005F;{i}=
&#x00A0;<br />&#x005C;begin{cases}&#x00A0;1&#x00A0;&#x0026;&#x00A0;&#x005C;text{if&#x00A0;$i&#x00A0;&#x005C;in&#x00A0;&#x005C;Lambda&#x005F;{k}$},&#x005C;&#x005C;
&#x00A0;<br />0&#x00A0;&#x0026;&#x005C;text{otherwise.}&#x00A0;&#x005C;end{cases}
&#x00A0;<br />&#x005C;]
</div>
</td></tr></table>
   <div class="newtheorem">
<!--l. 611--><p class="noindent"><span class="head">
<a 
  name="x1-6018r2"></a>
<span 
class="cmbx-10">Lemma 6.2.</span> </span>                                   <span 
class="cmti-10">Let                                     </span><!--l. 612--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></math>
<span 
class="cmti-10">be                 an                 arrangement.                 Then                 </span><!--l. 613--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
>
                   <mi 
>&#x03C7;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->B<!--/span--></mi><mo 
class="MathClass-rel">&#x2286;</mo><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow></msub 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
><!--span 
class="htf-calligraphy"-->B<!--/span--></mi></mfenced></mrow></msup 
><msup><mrow 
><mi 
>t</mi></mrow><mrow 
><mo 
>dim</mo><!--nolimits--> <mi 
>T</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->B<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mo 
class="MathClass-punc">.</mo>
</mrow></math>
</p>
   </div>
<!--l. 617--><p class="indent">   In order to compute <!--l. 617--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></math>
recall the definition of <!--l. 618--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>S</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></math>
from Lemma&#x00A0;<a 
href="#x1-3001r1">3.1<!--tex4ht:ref: lem-per--></a>. Since <!--l. 618--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>H</mi> <mo 
class="MathClass-rel">&#x2208;</mo><mi 
><!--span 
class="htf-calligraphy"-->B<!--/span--></mi></math>,
<!--l. 619--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">     <msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mi 
>H</mi></mrow></msub 
> <mo 
class="MathClass-rel">&#x2286;</mo><mi 
><!--span 
class="htf-calligraphy"-->B<!--/span--></mi></math>. Thus
if <!--l. 619--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>T</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->B<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>Y</mi> </math>
then <!--l. 620--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
><!--span 
class="htf-calligraphy"-->B<!--/span--></mi><mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>S</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>H</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></math>.
Let <!--l. 620--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msup><mrow 
><mi 
>L</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
> <mo 
class="MathClass-rel">=</mo> <mi 
>L</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>.
Then </p><table class="equation"><tr><td>

<!--l. 632--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>H</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
><!--span 
class="htf-calligraphy"-->B<!--/span--></mi><mo 
class="MathClass-rel">&#x2286;</mo><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow></msub 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
><!--span 
class="htf-calligraphy"-->B<!--/span--></mi></mfenced></mrow></msup 
><msup><mrow 
><mi 
>t</mi></mrow><mrow 
><mo 
>dim</mo><!--nolimits--> <mi 
>T</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->B<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>      <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>Y</mi> <mo 
class="MathClass-rel">&#x2208;</mo><msup><mrow 
><mi 
>L</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow></msub 
><msub><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
        </mrow><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->B<!--/span--></mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>S</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>H</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
><!--span 
class="htf-calligraphy"-->B<!--/span--></mi></mfenced></mrow></msup 
><msup><mrow 
><mi 
>t</mi></mrow><mrow 
><mo 
>dim</mo><!--nolimits--> <mi 
>Y</mi> </mrow></msup 
>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>     <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>Y</mi> <mo 
class="MathClass-rel">&#x2208;</mo><msup><mrow 
><mi 
>L</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow></msub 
><msub><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
        </mrow><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->B<!--/span--></mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>S</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>H</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
><!--span 
class="htf-calligraphy"-->B<!--/span--></mi><mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mi 
>H</mi></mrow></msub 
>
         </mfenced>
         </mrow></msup 
><msup><mrow 
><mi 
>t</mi></mrow><mrow 
><mo 
>dim</mo><!--nolimits--> <mi 
>Y</mi> </mrow></msup 
>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>     <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>Y</mi> <mo 
class="MathClass-rel">&#x2208;</mo><msup><mrow 
><mi 
>L</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow></msub 
><mi 
>&#x03BC;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>H</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow><msup><mrow 
><mi 
>t</mi></mrow><mrow 
><mo 
>dim</mo><!--nolimits--> <mi 
>Y</mi> </mrow></msup 
>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>     <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mo 
class="MathClass-bin">&#x2212;</mo><mi 
>&#x03C7;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">.</mo>
    </mtd></mtr></mtable>                                                                                    <mstyle 
    class="label" id="x1-6019r25"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(25)</td></tr></table>
   <div class="newtheorem">
<!--l. 634--><p class="noindent"><span class="head">
<a 
  name="x1-6020r3"></a>
<span 
class="cmbx-10">Corollary 6.3.</span> </span>                                  <span 
class="cmti-10">Let                                    </span><!--l. 635--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
<span 
class="cmti-10">be          a          triple          of          arrangements.          Then          </span><!--l. 636--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">

<mrow 
>
                     <mi 
>&#x03C0;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>&#x03C0;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">+</mo> <mi 
>t</mi><mi 
>&#x03C0;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">.</mo>
</mrow></math>
</p>
   </div>
   <div class="newtheorem">
<!--l. 639--><p class="noindent"><span class="head">
<a 
  name="x1-6021r2"></a>
<span 
class="cmbx-10">Definition 6.2.</span> </span>Let <!--l. 640--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
be a triple with respect to the hyperplane <!--l. 641--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>H</mi> <mo 
class="MathClass-rel">&#x2208;</mo><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></math>.
Call <!--l. 641--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>H</mi></math>
a <span 
class="cmti-10">separator </span>if <!--l. 642--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>T</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">/</mo><mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>L</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>.
</p>
   </div>
   <div class="newtheorem">
<!--l. 645--><p class="noindent"><span class="head">
<a 
  name="x1-6022r4"></a>
<span 
class="cmbx-10">Corollary 6.4.</span> </span> <span 
class="cmti-10">Let </span><!--l. 646--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math> <span 
class="cmti-10">be</span>
<span 
class="cmti-10">a triple with respect to </span><!--l. 646--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>H</mi> <mo 
class="MathClass-rel">&#x2208;</mo><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></math><span 
class="cmti-10">.</span>
     </p><ol type="1" class="enumerate1" start="1" 
>
     <li class="enumerate"><a 
  name="x1-6024x1"></a><span 
class="cmti-10">If </span><!--l. 650--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>H</mi></math>

     <span 
class="cmti-10">is a separator then </span><!--l. 651--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">      <mrow 
>
                                      <mi 
>&#x03BC;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mo 
class="MathClass-bin">&#x2212;</mo><mi 
>&#x03BC;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow>
</mrow></math>
     <span 
class="cmti-10">and hence </span><!--l. 653--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">      <mrow 
>
                                       <mfenced separators="" 
open="|"  close="|" ><mi 
>&#x03BC;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced> <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>&#x03BC;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced><mo 
class="MathClass-punc">.</mo>
</mrow></math>
     </li>
     <li class="enumerate"><a 
  name="x1-6026x2"></a><span 
class="cmti-10">If </span><!--l. 655--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>H</mi></math>
     <span 
class="cmti-10">is not a separator then </span><!--l. 656--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">      <mrow 
>
                                  <mi 
>&#x03BC;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>&#x03BC;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>&#x03BC;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow>
</mrow></math>

     <span 
class="cmti-10">and </span><!--l. 658--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">      <mrow 
>
                                 <mfenced separators="" 
open="|"  close="|" ><mi 
>&#x03BC;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced> <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>&#x03BC;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced> <mo 
class="MathClass-bin">+</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>&#x03BC;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced><mo 
class="MathClass-punc">.</mo>
</mrow></math></li></ol>
   </div>
<!--l. 662--><p class="indent">
</p>
   <div class="proof">
<!--l. 663--><p class="indent">   <span class="head">
<span 
class="cmti-10">Proof.</span> </span>It follows from Theorem&#x00A0;<a 
href="#x1-5002r1">5.1<!--tex4ht:ref: th-info-ow-ow--></a> that <!--l. 663--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>&#x03C0;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
has leading term <!--l. 665--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">      <mrow 
>
                                       <msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mi 
>r</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mi 
>&#x03BC;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msup><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>r</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mo 
class="MathClass-punc">.</mo>
</mrow></math>
The conclusion follows by comparing coefficients of the leading terms on both
sides of the equation in Corollary&#x00A0;<a 
href="#x1-6020r3">6.3<!--tex4ht:ref: tripleA--></a>. If <!--l. 669--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>H</mi></math>
is a separator then <!--l. 670--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>r</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x003C;</mo> <mi 
>r</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
and there is no contribution from <!--l. 671--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>&#x03C0;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>.
                                                                  &#x25A1;
</p>
   </div>
<!--l. 674--><p class="indent">   The Poincar&#x00E9; polynomial of an arrangement will appear repeatedly
in these notes. It will be shown to equal the Poincar&#x00E9; polynomial
of the graded algebras which we are going to associate with <!--l. 679--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">

<mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></math>.
It is also the Poincar&#x00E9; polynomial of the complement <!--l. 680--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>M</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math> for
a complex arrangement. Here we prove that the Poincar&#x00E9; polynomial is
the chamber counting function for a real arrangement. The complement <!--l. 684--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>M</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math> is a disjoint union
of chambers <!--l. 685--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">        <mrow 
>
                                    <mi 
>M</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x22C3;</mo>
  </mrow><mrow 
><mi 
>C</mi><mo 
class="MathClass-rel">&#x2208;</mo><mo 
class="MathClass-op">Cham</mo><!--nolimits--><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><mi 
>C</mi><mo 
class="MathClass-punc">.</mo>
</mrow></math>
The number of chambers is determined by the Poincar&#x00E9; polynomial as
follows.
</p>
   <div class="newtheorem">
<!--l. 690--><p class="noindent"><span class="head">
<a 
  name="x1-6027r5"></a>
<span 
class="cmbx-10">Theorem 6.5.</span> </span>                                  <span 
class="cmti-10">Let                                    </span><!--l. 691--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mi 
>R</mi></mrow></msub 
></math>
<span 
class="cmti-10">be             a             real             arrangement.             Then             </span><!--l. 692--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
>
                       <mfenced separators="" 
open="|"  close="|" ><mo 
class="MathClass-op">Cham</mo><!--nolimits--><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mi 
>R</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced> <mo 
class="MathClass-rel">=</mo> <mi 
>&#x03C0;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mi 
>R</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">.</mo>
</mrow></math>
</p>
   </div>
<!--l. 695--><p class="indent">

</p>
   <div class="proof">
<!--l. 696--><p class="indent">   <span class="head">
<span 
class="cmti-10">Proof.</span> </span>We check the properties required in Corollary&#x00A0;<a 
href="#x1-6022r4">6.4<!--tex4ht:ref: nsep--></a>: (i) follows from <!--l. 697--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>&#x03C0;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>&#x03A6;</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mn>1</mn></math>,
and (ii) is a consequence of Corollary&#x00A0;<a 
href="#x1-3011r4">3.4<!--tex4ht:ref: BI--></a>.                                                  &#x25A1;
</p>
   </div>
   <hr class="figure" /><div align="center" class="figure" 
><table class="figure"><tr class="figure"><td class="figure" 
>

<a 
  name="x1-60281"></a>

<br />                             <div align="center" class="caption"><table class="caption" 
><tr valign="baseline" class="caption"><td class="id">Figure&#x00A0;1:                                              </td><td  
class="content"><!--l. 703--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>Q</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mn>1</mn></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>x</mi><mi 
>y</mi><mi 
>z</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math></td></tr></table></div><!--tex4ht:label?: x1-60281-->

   </td></tr></table></div><hr class="endfigure" />
   <hr class="figure" /><div align="center" class="figure" 
><table class="figure"><tr class="figure"><td class="figure" 
>

<a 
  name="x1-60292"></a>

<br />                             <div align="center" class="caption"><table class="caption" 
><tr valign="baseline" class="caption"><td class="id">Figure&#x00A0;2:                                              </td><td  
class="content"><!--l. 708--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>Q</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->A<!--/span--></mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>x</mi><mi 
>y</mi><mi 
>z</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>y</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>y</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math></td></tr></table></div><!--tex4ht:label?: x1-60292-->

   </td></tr></table></div><hr class="endfigure" />
   <div class="newtheorem">
<!--l. 712--><p class="noindent"><span class="head">
<a 
  name="x1-6030r6"></a>
<span 
class="cmbx-10">Theorem 6.6.</span> </span> <span 
class="cmti-10">Let </span><!--l. 714--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>&#x03C6;</mi></math>
<span 
class="cmti-10">be a protocol for a random pair </span><!--l. 714--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></math><span 
class="cmti-10">.</span>
<span 
class="cmti-10">If one of </span><!--l. 715--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>&#x03C6;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
<span 
class="cmti-10">and </span><!--l. 715--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>&#x03C6;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
<span 
class="cmti-10">is a prefix of the other and </span><!--l. 716--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>X</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow></msub 
></math><span 
class="cmti-10">,</span>
<span 
class="cmti-10">then </span><!--l. 717--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">      <mrow 
>
                <msubsup><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow><mrow 
>
<mi 
>j</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>&#x221E;</mi></mrow></msubsup 
> <mo 
class="MathClass-rel">=</mo> <msubsup><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
>
<mi 
>j</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>&#x221E;</mi></mrow></msubsup 
> <mo 
class="MathClass-rel">=</mo> <msubsup><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
>
<mi 
>j</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow><mrow 
>
<mi 
>j</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>&#x221E;</mi></mrow></msubsup 
><mo 
class="MathClass-punc">.</mo>
</mrow></math>
</p>
   </div>
<!--l. 723--><p class="indent">
</p>
   <div class="proof">
<!--l. 724--><p class="indent">   <span class="head">
<span 
class="cmti-10">Proof.</span> </span>We show by induction on <!--l. 724--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>i</mi></math>
that <!--l. 725--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">      <mrow 
>
                    <msubsup><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow><mrow 
>
<mi 
>j</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>i</mi></mrow></msubsup 
> <mo 
class="MathClass-rel">=</mo> <msubsup><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
>
<mi 
>j</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>i</mi></mrow></msubsup 
> <mo 
class="MathClass-rel">=</mo> <msubsup><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
>
<mi 
>j</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow><mrow 
>
<mi 
>j</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>i</mi></mrow></msubsup 
><mo 
class="MathClass-punc">.</mo>
</mrow></math>

The induction hypothesis holds vacuously for <!--l. 730--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>i</mi> <mo 
class="MathClass-rel">=</mo> <mn>0</mn></math>.
Assume it holds for <!--l. 731--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>i</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>1</mn></math>,
in particular <!--l. 732--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msubsup><mrow 
><mrow><mo 
class="MathClass-open">[</mo><mrow><msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">]</mo></mrow></mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msubsup 
> <mo 
class="MathClass-rel">=</mo> <msubsup><mrow 
><mrow><mo 
class="MathClass-open">[</mo><mrow><msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">]</mo></mrow></mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msubsup 
></math>.
Then one of <!--l. 733--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msubsup><mrow 
><mrow><mo 
class="MathClass-open">[</mo><mrow><msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">]</mo></mrow></mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">=</mo><mi 
>i</mi></mrow><mrow 
><mi 
>&#x221E;</mi></mrow></msubsup 
></math>
and <!--l. 733--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msubsup><mrow 
><mrow><mo 
class="MathClass-open">[</mo><mrow><msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">]</mo></mrow></mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">=</mo><mi 
>i</mi></mrow><mrow 
><mi 
>&#x221E;</mi></mrow></msubsup 
></math>
is a prefix of the other which implies that one of <!--l. 734--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
and <!--l. 735--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
is a prefix of the other. If the <!--l. 735--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>i</mi></math>th
message is transmitted by <!--l. 736--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>P</mi></mrow><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->X<!--/span--></mi></mrow></msub 
></math>
then, by the separate-transmissions property and the induction hypothesis, <!--l. 737--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>,
hence one of <!--l. 738--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
and <!--l. 738--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
is a prefix of the other. By the implicit-termination property, neither <!--l. 739--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
nor <!--l. 739--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
can be a proper prefix of the other, hence they must be the same and <!--l. 741--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>.
If the <!--l. 741--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>i</mi></math>th
message is transmitted by <!--l. 742--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>P</mi></mrow><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->Y<!--/span--></mi></mrow></msub 
></math>
then, symmetrically, <!--l. 742--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
by the induction hypothesis and the separate-transmissions property, and, then, <!--l. 744--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
by the implicit-termination property, proving the induction step.                  &#x25A1;
</p>
   </div>
<!--l. 748--><p class="indent">   If <!--l. 748--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>&#x03C6;</mi></math> is a
protocol for <!--l. 748--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></math>,
and <!--l. 748--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>, <!--l. 748--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math> are distinct inputs in <!--l. 749--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>X</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow></msub 
></math>, then, by the correct-decision
property, <!--l. 750--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msubsup><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>&#x221E;</mi></mrow></msubsup 
><mo 
class="MathClass-rel">&#x2260;</mo><msubsup><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>&#x221E;</mi></mrow></msubsup 
></math>.
</p><!--l. 753--><p class="indent">   Equation&#x00A0;(<a 
href="#x1-6019r25">25<!--tex4ht:ref: E_SXgYy--></a>) defined <!--l. 753--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>P</mi></mrow><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->Y<!--/span--></mi></mrow></msub 
></math>&#x2019;s
ambiguity set <!--l. 753--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>X</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>Y</mi> </mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math> to be the
set of possible <!--l. 754--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>X</mi></math> values
when <!--l. 754--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>Y</mi> <mo 
class="MathClass-rel">=</mo> <mi 
>y</mi></math>. The last corollary
implies that for all <!--l. 755--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>y</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>Y</mi> </mrow></msub 
></math>,
the multiset<a 
href="testmath2.xml" name="testmath2.xml" ><sup>1</sup></a> of
codewords <!--l. 759--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mrow><mo 
class="MathClass-open">{</mo><mrow><msub><mrow 
><mi 
>&#x03C3;</mi></mrow><mrow 
><mi 
>&#x03C6;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-punc">:</mo> <mi 
>x</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>X</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>Y</mi> </mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">}</mo></mrow></math>
is prefix free.

</p>
   <h3 class="sectionHead"><span class="titlemark">7</span> <a 
  name="x1-70007"></a>One-Way Complexity</h3>
<!--l. 764--><p class="noindent"><!--l. 764--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">     <msub><mrow 
><mi 
>&#x0108;</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></math>,
the one-way complexity of a random pair <!--l. 764--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></math>, is the number of bits <!--l. 765--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>P</mi></mrow><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->X<!--/span--></mi></mrow></msub 
></math> must transmit in the
worst case when <!--l. 766--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>P</mi></mrow><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->Y<!--/span--></mi></mrow></msub 
></math>
is not permitted to transmit any feedback messages. Starting with <!--l. 767--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>X</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow></msub 
></math>, the support set of <!--l. 767--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></math>, we define <!--l. 767--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>G</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></math>, the <span 
class="cmti-10">characteristic</span>
<span 
class="cmti-10">hypergraph </span>of <!--l. 768--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></math>,
and show that <!--l. 769--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">       <mrow 
><msub><mrow 
>
                               <mi 
>&#x0108;</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">&#x2308;</mo><mrow><mo 
>log</mo><!--nolimits--><mi 
>&#x03C7;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>G</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">&#x2309;</mo></mrow><mspace class="nbsp" /><mo 
class="MathClass-punc">.</mo>
</mrow></math>
</p><!--l. 773--><p class="indent">   Let <!--l. 773--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></math> be a random
pair. For each <!--l. 773--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>y</mi></math> in <!--l. 773--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>Y</mi> </mrow></msub 
></math>, the support set of <!--l. 774--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>Y</mi> </math>, Equation&#x00A0;(<a 
href="#x1-6019r25">25<!--tex4ht:ref: E_SXgYy--></a>) defined <!--l. 774--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>X</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>Y</mi> </mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math> to be the set of possible <!--l. 775--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>x</mi></math> values when <!--l. 775--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>Y</mi> <mo 
class="MathClass-rel">=</mo> <mi 
>y</mi></math>. The <span 
class="cmti-10">characteristic</span>
<span 
class="cmti-10">hypergraph </span><!--l. 775--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>G</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></math>
of <!--l. 776--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>X</mi><mo 
class="MathClass-punc">,</mo><mi 
>Y</mi> </mrow><mo 
class="MathClass-close">)</mo></mrow></math> has <!--l. 776--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>X</mi></mrow></msub 
></math> as its vertex set and
the hyperedge <!--l. 776--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>X</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>Y</mi> </mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
for each <!--l. 777--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>y</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>Y</mi> </mrow></msub 
></math>.
</p><!--l. 780--><p class="indent">   We can now prove a continuity theorem. </p><div class="newtheorem">
<!--l. 781--><p class="noindent"><span class="head">
<a 
  name="x1-7001r1"></a>
<span 
class="cmbx-10">Theorem 7.1.</span> </span> <span 
class="cmti-10">Let </span><!--l. 782--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>&#x03A9;</mi> <mo 
class="MathClass-rel">&#x2282;</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
></math>
<span 
class="cmti-10">be an open set, let </span><!--l. 783--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>u</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>B</mi><mi 
>V</mi> <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">;</mo><msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>m</mi></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math><span 
class="cmti-10">,</span>

<span 
class="cmti-10">and let</span> </p><table class="equation"><tr><td>
<!--l. 788--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">      
<msubsup><mrow 
><mi 
>T</mi></mrow><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>u</mi></mrow></msubsup 
> <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="{"  close="}" ><mi 
>y</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>m</mi></mrow></msup 
> <mo 
class="MathClass-punc">:</mo> <mi 
>y</mi> <mo 
class="MathClass-rel">=</mo> <mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">+</mo> <mfenced separators="" 
open="&#x3008;"  close="&#x3009;" > <mfrac><mrow 
><mi 
>D</mi><mi 
>u</mi></mrow>
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mi 
>D</mi><mi 
>u</mi></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo><mi 
>z</mi></mfenced><!--mstyle 
class="text"--><mtext >&#x000A0;for&#x000A0;some&#x000A0;</mtext><!--/mstyle--><mi 
>z</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
></mfenced> <mstyle 
    class="label" id="x1-7002r26"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(26)</td></tr></table>
<span 
class="cmti-10">for every </span><!--l. 789--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>x</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>&#x03A9;</mi><mo 
class="MathClass-bin">&#x2216;</mo><msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>u</mi></mrow></msub 
></math><span 
class="cmti-10">. Let </span><!--l. 789--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>f</mi> <mo 
class="MathClass-punc">:</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>m</mi></mrow></msup 
> <mo 
class="MathClass-rel">&#x2192;</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>k</mi></mrow></msup 
></math> <span 
class="cmti-10">be a Lipschitz continuous</span>
<span 
class="cmti-10">function such that </span><!--l. 790--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>0</mn></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mn>0</mn></math><span 
class="cmti-10">,</span>
<span 
class="cmti-10">and let </span><!--l. 791--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>v</mi> <mo 
class="MathClass-rel">=</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>u</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">:</mo> <mi 
>&#x03A9;</mi> <mo 
class="MathClass-rel">&#x2192;</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>k</mi></mrow></msup 
></math><span 
class="cmti-10">.</span>
<span 
class="cmti-10">Then </span><!--l. 791--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>v</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>B</mi><mi 
>V</mi> <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">;</mo><msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>k</mi></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
<span 
class="cmti-10">and </span><table class="equation"><tr><td>

<!--l. 796--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">      
<mi 
>J</mi><mi 
>v</mi> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mfenced separators="" 
open=""  close="|" ><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2297;</mo> <msub><mrow 
><mi 
>&#x03BD;</mi></mrow><mrow 
>
<mi 
>u</mi></mrow></msub 
> <mo 
class="MathClass-punc">&#x22C5;</mo> <msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->H<!--/span--></mi></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msub 
></mfenced> </mrow><mrow 
><msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>u</mi></mrow></msub 
></mrow></msub 
><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-7003r27"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(27)</td></tr></table>
<span 
class="cmti-10">In addition, for </span><!--l. 797--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></math><span 
class="cmti-10">-almost</span>
<span 
class="cmti-10">every </span><!--l. 797--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>x</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>&#x03A9;</mi></math> <span 
class="cmti-10">the restriction</span>
<span 
class="cmti-10">of the function </span><!--l. 798--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>f</mi></math> <span 
class="cmti-10">to </span><!--l. 798--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msubsup><mrow 
><mi 
>T</mi></mrow><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>u</mi></mrow></msubsup 
></math> <span 
class="cmti-10">is differentiable</span>
<span 
class="cmti-10">at </span><!--l. 798--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
<span 
class="cmti-10">and </span><table class="equation"><tr><td>

<!--l. 802--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">      
<mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>v</mi> <mo 
class="MathClass-rel">=</mo> <mi 
>&#x2207;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mfenced separators="" 
open=""  close="|" ><mi 
>f</mi></mfenced></mrow><mrow 
><msubsup><mrow 
><mi 
>T</mi></mrow><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>u</mi></mrow></msubsup 
></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mfrac><mrow 
><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mrow>
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac> <mo 
class="MathClass-punc">&#x22C5;</mo><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-7004r28"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(28)</td></tr></table>
   </div>
<!--l. 805--><p class="indent">   Before proving the theorem, we state without proof three elementary remarks
which will be useful in the sequel. </p><div class="newtheorem">
<!--l. 807--><p class="noindent"><span class="head">
<a 
  name="x1-7005r1"></a>
<span 
class="cmti-10">Remark 7.1.</span> </span> Let <!--l. 808--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>&#x03C9;</mi><mo 
class="MathClass-punc">:</mo>  <mfenced separators="" 
open="]"  close="[" ><mn>0</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-bin">+</mo><mi 
>&#x221E;</mi></mfenced><mo 
class="MathClass-rel">&#x2192;</mo> <mfenced separators="" 
open="]"  close="[" ><mn>0</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-bin">+</mo><mi 
>&#x221E;</mi></mfenced></math>
be a continuous function such that <!--l. 809--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>&#x03C9;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2192;</mo> <mn>0</mn></math>
as <!--l. 809--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>t</mi> <mo 
class="MathClass-rel">&#x2192;</mo> <mn>0</mn></math>.
Then <!--l. 811--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">      <mrow 
>
                        <msub><mrow 
><mo 
>lim</mo></mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-rel">&#x2192;</mo><msup><mrow 
><mn>0</mn></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
></mrow></msub 
><mi 
>g</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03C9;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>h</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>L</mi> <mo 
class="MathClass-rel">&#x21D4;</mo><msub><mrow 
><mo 
> lim</mo></mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-rel">&#x2192;</mo><msup><mrow 
><mn>0</mn></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
></mrow></msub 
><mi 
>g</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>h</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>L</mi>
</mrow></math>
for any function <!--l. 813--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>g</mi><mo 
class="MathClass-punc">:</mo>  <mfenced separators="" 
open="]"  close="[" ><mn>0</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-bin">+</mo><mi 
>&#x221E;</mi></mfenced><mo 
class="MathClass-rel">&#x2192;</mo> <mi 
>R</mi></math>.
</p>
   </div>

   <div class="newtheorem">
<!--l. 815--><p class="noindent"><span class="head">
<a 
  name="x1-7006r2"></a>
<span 
class="cmti-10">Remark 7.2.</span> </span> Let <!--l. 816--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>g</mi><mo 
class="MathClass-punc">:</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
> <mo 
class="MathClass-rel">&#x2192;</mo> <mi 
>R</mi></math>
be a Lipschitz continuous function and assume that <!--l. 818--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">      <mrow 
>
                                 <mi 
>L</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
><mo 
> lim</mo></mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-rel">&#x2192;</mo><msup><mrow 
><mn>0</mn></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
></mrow></msub 
> <mfrac><mrow 
><mi 
>g</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>h</mi><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>g</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>0</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
       <mrow 
><mi 
>h</mi></mrow></mfrac>
</mrow></math>
exists for every <!--l. 819--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>z</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msup><mrow 
><mi 
>Q</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
></math>
and that <!--l. 819--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>L</mi></math>
is a linear function of <!--l. 820--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>z</mi></math>.
Then <!--l. 820--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>g</mi></math>
is differentiable at 0.
</p>
   </div>
   <div class="newtheorem">
<!--l. 822--><p class="noindent"><span class="head">
<a 
  name="x1-7007r3"></a>
<span 
class="cmti-10">Remark 7.3.</span> </span>                                      Let                                         <!--l. 823--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>A</mi><mo 
class="MathClass-punc">:</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
> <mo 
class="MathClass-rel">&#x2192;</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>m</mi></mrow></msup 
></math>
be            a            linear            function,            and            let            <!--l. 824--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>f</mi> <mo 
class="MathClass-punc">:</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>m</mi></mrow></msup 
> <mo 
class="MathClass-rel">&#x2192;</mo> <mi 
>R</mi></math>
be         a         function.         Then         the         restriction         of         <!--l. 825--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>f</mi></math>
to                      the                      range                      of                      <!--l. 825--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>A</mi></math>
is        differentiable        at        0        if        and        only        if        <!--l. 826--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>A</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">:</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
> <mo 
class="MathClass-rel">&#x2192;</mo> <mi 
>R</mi></math>
is               differentiable               at               0               and               <!--l. 828--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">

<mrow 
>
                    <mi 
>&#x2207;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mfenced separators="" 
open=""  close="|" ><mi 
>f</mi></mfenced></mrow><mrow 
><mo 
class="MathClass-op">Im</mo><!--nolimits--><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>A</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>0</mn></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>A</mi> <mo 
class="MathClass-rel">=</mo> <mi 
>&#x2207;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>A</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>0</mn></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">.</mo>
</mrow></math>
</p>
   </div>
<!--l. 831--><p class="indent">
</p>
   <div class="proof">
<!--l. 832--><p class="indent">   <span class="head">
<span 
class="cmti-10">Proof.</span> </span>We begin by showing that <!--l. 832--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>v</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>B</mi><mi 
>V</mi> <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">;</mo><msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>k</mi></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
and </p><table class="equation"><tr><td>
<!--l. 835--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
 <mfenced separators="" 
open="|"  close="|" ><mi 
>D</mi><mi 
>v</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>B</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2264;</mo> <mi 
>K</mi> <mfenced separators="" 
open="|"  close="|" ><mi 
>D</mi><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>B</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/><mi 
>&#x2200;</mi><mi 
>B</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>B</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo><mstyle 
    class="label" id="x1-7008r29"  ></mstyle><!--endlabel-->
</math>

</td><td class="eq-no">(29)</td></tr></table>
where <!--l. 836--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>K</mi> <mo 
class="MathClass-rel">&#x003E;</mo> <mn>0</mn></math> is the Lipschitz
constant of <!--l. 836--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>f</mi></math>.
By (<a 
href="#x1-3015r13">13<!--tex4ht:ref: sum-Di--></a>) and by the approximation result quoted in <span 
class="cmsy-10">&#x00A7;</span><a 
href="#x1-30003">3<!--tex4ht:ref: s:mt--></a>, it is possible to find a sequence <!--l. 838--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>h</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2282;</mo> <msup><mrow 
><mi 
>C</mi></mrow><mrow 
><mn>1</mn></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">;</mo><msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>m</mi></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math> converging
to <!--l. 838--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>u</mi></math> in <!--l. 839--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msup><mrow 
><mi 
>L</mi></mrow><mrow 
><mn>1</mn></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">;</mo><msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>m</mi></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math> and such that <!--l. 840--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
>
                   <msub><mrow 
><mo 
>lim</mo></mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-rel">&#x2192;</mo><mo 
class="MathClass-bin">+</mo><mi 
>&#x221E;</mi></mrow></msub 
><msub><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
        </mrow><mrow 
><mi 
>&#x03A9;</mi></mrow></msub 
> <mfenced separators="" 
open="|"  close="|" ><mi 
>&#x2207;</mi><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>h</mi></mrow></msub 
></mfenced> <mi 
>d</mi><mi 
>x</mi> <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>D</mi><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">.</mo>
</mrow></math> The functions <!--l. 841--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>h</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>h</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math> are locally Lipschitz
continuous in <!--l. 841--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>&#x03A9;</mi></math>,
and the definition of differential implies that <!--l. 842--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mfenced separators="" 
open="|"  close="|" ><mi 
>&#x2207;</mi><msub><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>h</mi></mrow></msub 
></mfenced> <mo 
class="MathClass-rel">&#x2264;</mo> <mi 
>K</mi> <mfenced separators="" 
open="|"  close="|" ><mi 
>&#x2207;</mi><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>h</mi></mrow></msub 
></mfenced></math> almost everywhere
in <!--l. 843--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>&#x03A9;</mi></math>.
The lower semicontinuity of the total variation and (<a 
href="#x1-3015r13">13<!--tex4ht:ref: sum-Di--></a>) yield <table class="equation"><tr><td>

<!--l. 851--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>  <mfenced separators="" 
open="|"  close="|" ><mi 
>D</mi><mi 
>v</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2264;</mo><msub><mrow 
><mo 
> liminf</mo> </mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-rel">&#x2192;</mo><mo 
class="MathClass-bin">+</mo><mi 
>&#x221E;</mi></mrow></msub 
><mfenced separators="" 
open="|"  close="|" ><mi 
>D</mi><msub><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>h</mi></mrow></msub 
></mfenced> <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
><mo 
> liminf</mo> </mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-rel">&#x2192;</mo><mo 
class="MathClass-bin">+</mo><mi 
>&#x221E;</mi></mrow></msub 
><msub><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
          </mrow><mrow 
><mi 
>&#x03A9;</mi></mrow></msub 
> <mfenced separators="" 
open="|"  close="|" ><mi 
>&#x2207;</mi><msub><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>h</mi></mrow></msub 
></mfenced> <mi 
>d</mi><mi 
>x</mi>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>                                      <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">&#x2264;</mo> <mi 
>K</mi><msub><mrow 
><mo 
>liminf</mo> </mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-rel">&#x2192;</mo><mo 
class="MathClass-bin">+</mo><mi 
>&#x221E;</mi></mrow></msub 
><msub><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
          </mrow><mrow 
><mi 
>&#x03A9;</mi></mrow></msub 
> <mfenced separators="" 
open="|"  close="|" ><mi 
>&#x2207;</mi><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>h</mi></mrow></msub 
></mfenced> <mi 
>d</mi><mi 
>x</mi> <mo 
class="MathClass-rel">=</mo> <mi 
>K</mi> <mfenced separators="" 
open="|"  close="|" ><mi 
>D</mi><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">.</mo>
    </mtd></mtr></mtable>                                                                                    <mstyle 
    class="label" id="x1-7009r30"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(30)</td></tr></table>
Since <!--l. 852--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>0</mn></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mn>0</mn></math>, we
have also <!--l. 853--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">        <mrow 
><msub><mrow 
>
                                   <mo 
class="MathClass-op">&#x222B;</mo>
                  </mrow><mrow 
><mi 
>&#x03A9;</mi></mrow></msub 
> <mfenced separators="" 
open="|"  close="|" ><mi 
>v</mi></mfenced> <mi 
>d</mi><mi 
>x</mi> <mo 
class="MathClass-rel">&#x2264;</mo> <mi 
>K</mi><msub><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>&#x03A9;</mi></mrow></msub 
> <mfenced separators="" 
open="|"  close="|" ><mi 
>u</mi></mfenced> <mi 
>d</mi><mi 
>x</mi><mo 
class="MathClass-punc">;</mo>
</mrow></math>
therefore <!--l. 854--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>u</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>B</mi><mi 
>V</mi> <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">;</mo><msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>k</mi></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math>.
Repeating the same argument for every open set <!--l. 855--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>A</mi> <mo 
class="MathClass-rel">&#x2282;</mo> <mi 
>&#x03A9;</mi></math>, we get (<a 
href="#x1-7008r29">29<!--tex4ht:ref: e:bomb--></a>) for
every <!--l. 856--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>B</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>B</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>,
because <!--l. 856--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mfenced separators="" 
open="|"  close="|" ><mi 
>D</mi><mi 
>v</mi></mfenced></math>,
<!--l. 856--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">     <mfenced separators="" 
open="|"  close="|" ><mi 
>D</mi><mi 
>u</mi></mfenced></math>
are Radon measures. To prove Lemma&#x00A0;<a 
href="#x1-6002r1">6.1<!--tex4ht:ref: limbog--></a>, first we observe that <table class="equation"><tr><td>

<!--l. 860--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>v</mi></mrow></msub 
> <mo 
class="MathClass-rel">&#x2282;</mo> <msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>u</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mspace width="2em" class="qquad"/><mi 
>&#x1E7D;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/><mi 
>&#x2200;</mi><mi 
>x</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>&#x03A9;</mi><mo 
class="MathClass-bin">&#x2216;</mo><msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>u</mi></mrow></msub 
><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-7010r31"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(31)</td></tr></table>
In fact, for every <!--l. 861--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>&#x025B;</mi> <mo 
class="MathClass-rel">&#x003E;</mo> <mn>0</mn></math>
we have <!--l. 862--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">       <mrow 
>
    <mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>y</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>B</mi></mrow><mrow 
><mi 
>&#x03C1;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-punc">:</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>v</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced> <mo 
class="MathClass-rel">&#x003E;</mo> <mi 
>&#x025B;</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mo 
class="MathClass-rel">&#x2282;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>y</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>B</mi></mrow><mrow 
><mi 
>&#x03C1;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-punc">:</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>u</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced> <mo 
class="MathClass-rel">&#x003E;</mo> <mi 
>&#x025B;</mi><mo 
class="MathClass-bin">/</mo><mi 
>K</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mo 
class="MathClass-punc">,</mo>
</mrow></math> hence

<!--l. 865--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     <mrow 
>
                   <msub><mrow 
><mo 
>lim</mo></mrow><mrow 
><mi 
>&#x03C1;</mi><mo 
class="MathClass-rel">&#x2192;</mo><msup><mrow 
><mn>0</mn></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
></mrow></msub 
> <mfrac><mrow 
> <mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>y</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>B</mi></mrow><mrow 
><mi 
>&#x03C1;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-punc">:</mo> <mfenced separators="" 
open="|"  close="|" ><mi 
>v</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced> <mo 
class="MathClass-rel">&#x003E;</mo> <mi 
>&#x025B;</mi></mrow><mo 
class="MathClass-close">}</mo></mrow></mfenced></mrow> 
          <mrow 
><msup><mrow 
><mi 
>&#x03C1;</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
></mrow></mfrac>                        <mo 
class="MathClass-rel">=</mo> <mn>0</mn>
</mrow></math> whenever <!--l. 867--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>x</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>&#x03A9;</mi><mo 
class="MathClass-bin">&#x2216;</mo><msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>u</mi></mrow></msub 
></math>. By a similar argument, if <!--l. 867--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>x</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>u</mi></mrow></msub 
></math>is a point such that there
exists a triplet <!--l. 868--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo></mrow></msup 
><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>&#x03BD;</mi></mrow><mrow 
><mi 
>u</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></math> satisfying
(<a 
href="#x1-3016r14">14<!--tex4ht:ref: detK1--></a>), (<a 
href="#x1-3017r15">15<!--tex4ht:ref: detK2--></a>), then <!--l. 870--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">       <mrow 
>
            <mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>v</mi></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <msup><mrow 
><mi 
>v</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2297;</mo> <msub><mrow 
><mi 
>&#x03BD;</mi></mrow><mrow 
>
<mi 
>v</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2297;</mo> <msub><mrow 
><mi 
>&#x03BD;</mi></mrow><mrow 
>
<mi 
>u</mi></mrow></msub 
><!--mstyle 
class="text"--><mtext >if&#x000A0;</mtext><!--/mstyle--><mi 
>x</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>v</mi></mrow></msub 
>
</mrow></math>
and <!--l. 874--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
if <!--l. 874--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>x</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>u</mi></mrow></msub 
><mo 
class="MathClass-bin">&#x2216;</mo><msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>v</mi></mrow></msub 
></math>.
Hence, by (1.8) we get <table class="equation"><tr><td>

<!--l. 880--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow> <mi 
>J</mi><mi 
>v</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>B</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>B</mi><mo 
class="MathClass-bin">&#x2229;</mo><msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>v</mi></mrow></msub 
></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>v</mi></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msup><mrow 
><mi 
>v</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2297;</mo> <msub><mrow 
><mi 
>&#x03BD;</mi></mrow><mrow 
>
<mi 
>v</mi></mrow></msub 
> <mi 
>d</mi><msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->H<!--/span--></mi></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msub 
><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>B</mi><mo 
class="MathClass-bin">&#x2229;</mo><msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>v</mi></mrow></msub 
></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2297;</mo> <msub><mrow 
><mi 
>&#x03BD;</mi></mrow><mrow 
>
<mi 
>u</mi></mrow></msub 
> <mi 
>d</mi><msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->H<!--/span--></mi></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msub 
>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>                                                                 <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>B</mi><mo 
class="MathClass-bin">&#x2229;</mo><msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>u</mi></mrow></msub 
></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2297;</mo> <msub><mrow 
><mi 
>&#x03BD;</mi></mrow><mrow 
>
<mi 
>u</mi></mrow></msub 
> <mi 
>d</mi><msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->H<!--/span--></mi></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msub 
>
      </mtd></mtr></mtable>                                                                                                                            
</math>
</td></tr></table>
and Lemma&#x00A0;<a 
href="#x1-6002r1">6.1<!--tex4ht:ref: limbog--></a> is proved.                                                                       &#x25A1;
   </div>
<!--l. 884--><p class="indent">   To prove (<a 
href="#x1-7010r31">31<!--tex4ht:ref: e:SS--></a>), it is not restrictive to assume that <!--l. 884--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>k</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn></math>.
Moreover, to simplify our notation, from now on we shall assume that <!--l. 886--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>&#x03A9;</mi> <mo 
class="MathClass-rel">=</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
></math>. The proof of
(<a 
href="#x1-7010r31">31<!--tex4ht:ref: e:SS--></a>) is divided into two steps. In the first step we prove the statement in the one-dimensional
case <!--l. 888--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>n</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></math>,
using Theorem&#x00A0;<a 
href="#x1-5003r2">5.2<!--tex4ht:ref: th-weak-ske-owf--></a>. In the second step we achieve the general result using
Theorem&#x00A0;<a 
href="#x1-7001r1">7.1<!--tex4ht:ref: t:conl--></a>.
</p>
   <h4 class="likesubsectionHead"><a 
  name="x1-80007"></a>Step 1</h4> Assume that <!--l. 892--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>n</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn></math>.
Since <!--l. 892--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>u</mi></mrow></msub 
></math> is at most countable,
(<a 
href="#x1-3006r7">7<!--tex4ht:ref: sum-bij--></a>)yields that <!--l. 893--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>v</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>u</mi></mrow></msub 
><mo 
class="MathClass-bin">&#x2216;</mo><msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>v</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mn>0</mn></math>, so that
(<a 
href="#x1-4001r19">19<!--tex4ht:ref: e:st--></a>)and (<a 
href="#x1-4003r21">21<!--tex4ht:ref: e:barwq--></a>) imply that <!--l. 894--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>D</mi><mi 
>v</mi> <mo 
class="MathClass-rel">=</mo> <mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>v</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>J</mi><mi 
>v</mi></math>
is the Radon-Nikod&#x00FD;m decomposition of <!--l. 895--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>D</mi><mi 
>v</mi></math>
in absolutely continuous and singular part with respect to <!--l. 896--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></math>. By
Theorem&#x00A0;<a 
href="#x1-5003r2">5.2<!--tex4ht:ref: th-weak-ske-owf--></a>, we have <table class="equation"><tr><td>

<!--l. 905--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
 <mfrac><mrow 
><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>v</mi></mrow>
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
><mo 
> lim</mo></mrow><mrow 
><mi 
>s</mi><mo 
class="MathClass-rel">&#x2192;</mo><msup><mrow 
><mi 
>t</mi></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
></mrow></msub 
>  <mfrac><mrow 
><mi 
>D</mi><mi 
>v</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac><mo 
class="MathClass-punc">,</mo><mspace width="2em" class="qquad"/> <mfrac><mrow 
><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi></mrow> 
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
><mo 
> lim</mo></mrow><mrow 
><mi 
>s</mi><mo 
class="MathClass-rel">&#x2192;</mo><msup><mrow 
><mi 
>t</mi></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
></mrow></msub 
>  <mfrac><mrow 
><mi 
>D</mi><mi 
>u</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac>
</math>
</td></tr></table>
<!--l. 906--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">     <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></math>-almost
everywhere in <!--l. 906--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>R</mi></math>.
It is well known (see, for instance, <span class="cite">[<a 
href="#Xste:sint">12</a>, 2.5.16]</span>) that every one-dimensional function of bounded
variation <!--l. 908--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>w</mi></math>
has a unique left continuous representative, i.e., a function <!--l. 909--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>&#x0175;</mi></math> such that <!--l. 909--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>&#x0175;</mi> <mo 
class="MathClass-rel">=</mo> <mi 
>w</mi></math> almost everywhere
and <!--l. 910--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mo 
>lim</mo></mrow><mrow 
><mi 
>s</mi><mo 
class="MathClass-rel">&#x2192;</mo><msup><mrow 
><mi 
>t</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo></mrow></msup 
></mrow></msub 
><mi 
>&#x0175;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>&#x0175;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math> for
every <!--l. 910--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>t</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>R</mi></math>.
These conditions imply <table class="equation"><tr><td>

<!--l. 916--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>D</mi><mi 
>u</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="]"  close="[" ><mo 
class="MathClass-bin">&#x2212;</mo><mi 
>&#x221E;</mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo><mspace width="2em" class="qquad"/><mover 
accent="true"><mrow 
><mi 
>v</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>D</mi><mi 
>v</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="]"  close="[" ><mo 
class="MathClass-bin">&#x2212;</mo><mi 
>&#x221E;</mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/><mi 
>&#x2200;</mi><mi 
>t</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>R</mi><mstyle 
    class="label" id="x1-8001r32"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(32)</td></tr></table>
and <table class="equation"><tr><td>
<!--l. 919--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<mover 
accent="true"><mrow 
><mi 
>v</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/><mi 
>&#x2200;</mi><mi 
>t</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>R</mi><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-8002r33"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(33)</td></tr></table>

Let <!--l. 920--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>t</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>R</mi></math> be
such that <!--l. 921--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x003E;</mo> <mn>0</mn></math>
for every <!--l. 921--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>s</mi> <mo 
class="MathClass-rel">&#x003E;</mo> <mi 
>t</mi></math>
and assume that the limits in (<a 
href="#x1-4004r22">22<!--tex4ht:ref: joe--></a>) exist. By (<a 
href="#x1-4005r23">23<!--tex4ht:ref: j:mark--></a>) and (<a 
href="#x1-6017r24">24<!--tex4ht:ref: far-d--></a>) we get <table class="equation"><tr><td>
<!--l. 936--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow> <mfrac><mrow 
><mover 
accent="true"><mrow 
><mi 
>v</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo><mover 
accent="true"><mrow 
><mi 
>v</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac> <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mfrac><mrow 
><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
      <mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>                 <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mfrac><mrow 
><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">+</mo><mfrac><mrow> <mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mrow> 
<mrow><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
                    <mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>                 <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-bin">+</mo> <mfrac><mrow 
><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">+</mo><mfrac><mrow> <mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mrow> 
<mrow><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
                   <mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac>
    </mtd></mtr></mtable>                                                                                                   
</math>
</td></tr></table>
for every <!--l. 937--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>s</mi> <mo 
class="MathClass-rel">&#x003E;</mo> <mi 
>t</mi></math>. Using the
Lipschitz condition on <!--l. 937--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>f</mi></math>
we find

<!--l. 939--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     
<mtable 
class="multline-star">
<mtr><mtd 
class="multline-star"> <mfenced separators="" 
open="|"  close="|" ><mfrac><mrow 
><mover 
accent="true"><mrow 
><mi 
>v</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo><mover 
accent="true"><mrow 
><mi 
>v</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow>
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac> <mo 
class="MathClass-bin">&#x2212;</mo><mfrac><mrow 
><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">+</mo><mfrac><mrow> <mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mrow> 
<mrow><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
                   <mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac>                    </mfenced>
</mtd></mtr><mtr><mtd 
class="multline-star"> <mo 
class="MathClass-rel">&#x2264;</mo> <mi 
>K</mi> <mfenced separators="" 
open="|"  close="|" ><mfrac><mrow 
><mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo><mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow>
  <mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac>   <mo 
class="MathClass-bin">&#x2212;</mo> <mfrac><mrow 
><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mrow> 
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced><mo 
class="MathClass-punc">.</mo>                                                    </mtd></mtr></mtable>
</math>
<!--l. 949--><p class="nopar">
By (<a 
href="#x1-7008r29">29<!--tex4ht:ref: e:bomb--></a>), the function <!--l. 951--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mi 
>s</mi> <mo 
class="MathClass-rel">&#x2192;</mo> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><mrow><mo 
class="MathClass-open">(</mo><mrow><mrow class="mathinner"> <mfenced separators="" 
open="["  close="[" ><mi 
>t</mi><mo 
class="MathClass-punc">,</mo><mi 
>s</mi></mfenced></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></math> is
continuous and converges to 0 as <!--l. 953--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>s</mi> <mi 
>&#x2193;</mi> <mi 
>t</mi></math>.
Therefore Remark&#x00A0;<a 
href="#x1-7005r1">7.1<!--tex4ht:ref: r:omb--></a> and the previous inequality imply <!--l. 955--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
>
      <mfrac><mrow 
><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>v</mi></mrow>
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
><mo 
> lim</mo></mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-rel">&#x2192;</mo><msup><mrow 
><mn>0</mn></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow></msup 
></mrow></msub 
> <mfrac><mrow 
><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">+</mo> <mi 
>h</mi><mfrac><mrow> <mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi></mrow>
<mrow><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
                   <mrow 
><mi 
>h</mi></mrow></mfrac>                  <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi></mfenced><!--mstyle 
class="text"--><mtext >-a.e.&#x000A0;in&#x000A0;</mtext><!--/mstyle--><mi 
>R</mi><mo 
class="MathClass-punc">.</mo>
</mrow></math> By (<a 
href="#x1-4004r22">22<!--tex4ht:ref: joe--></a>), <!--l. 958--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>&#x00FB;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math> for every
<!--l. 959--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">     <mi 
>x</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>R</mi><mo 
class="MathClass-bin">&#x2216;</mo><msub><mrow 
><mi 
>S</mi></mrow><mrow 
><mi 
>u</mi></mrow></msub 
></math>;
moreover, applying the same argument to the functions <!--l. 960--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>u</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>, <!--l. 960--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msup><mrow 
><mi 
>v</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>v</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>, we get <!--l. 961--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">

<mrow 
>
       <mfrac><mrow 
><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>v</mi></mrow>
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
><mo 
> lim</mo></mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-rel">&#x2192;</mo><mn>0</mn></mrow></msub 
><mfrac><mrow 
><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">+</mo> <mi 
>h</mi><mfrac><mrow> <mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi></mrow>
<mrow><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
                   <mrow 
><mi 
>h</mi></mrow></mfrac>                 <mspace width="2em" class="qquad"/> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi></mfenced><!--mstyle 
class="text"--><mtext >-a.e.&#x000A0;in&#x000A0;</mtext><!--/mstyle--><mi 
>R</mi>
</mrow></math> and
our statement is proved.
</p>
   <h4 class="likesubsectionHead"><a 
  name="x1-90007"></a>Step 2</h4>
<!--l. 969--><p class="noindent">Let us consider now the general case <!--l. 969--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>n</mi> <mo 
class="MathClass-rel">&#x003E;</mo> <mn>1</mn></math>. Let <!--l. 969--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>&#x03BD;</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
></math> be such that <!--l. 970--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mfenced separators="" 
open="|"  close="|" ><mi 
>&#x03BD;</mi></mfenced> <mo 
class="MathClass-rel">=</mo> <mn>1</mn></math>, and let <!--l. 970--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>&#x03C0;</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>y</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
> <mo 
class="MathClass-punc">:</mo> <mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mi 
>y</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mn>0</mn></mrow><mo 
class="MathClass-close">}</mo></mrow></math>. In the following, we
shall identify <!--l. 971--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
></math> with <!--l. 972--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>&#x03C0;</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
> <mo 
class="MathClass-bin">&#x00D7;</mo> <mi 
>R</mi></math>, and we shall denote by <!--l. 972--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>y</mi></math> the variable ranging
in <!--l. 973--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>&#x03C0;</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></math> and by <!--l. 973--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>t</mi></math> the variable
ranging in <!--l. 973--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>R</mi></math>.
By the just proven one-dimensional result, and by Theorem&#x00A0;<a 
href="#x1-3008r3">3.3<!--tex4ht:ref: thm-main--></a>, we get <!--l. 975--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
>
<msub><mrow 
><mo 
>lim</mo></mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-rel">&#x2192;</mo><mn>0</mn></mrow></msub 
><mfrac><mrow 
><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>t</mi><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">+</mo> <mi 
>h</mi><mfrac><mrow> <mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mrow>
<mrow><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>t</mi><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow>

                           <mrow 
><mi 
>h</mi></mrow></mfrac>                  <mo 
class="MathClass-rel">=</mo>  <mfrac><mrow 
><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><msub><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mrow> 
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mfenced> </mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mfenced> <!--mstyle 
class="text"--><mtext >-a.e.&#x000A0;in&#x000A0;</mtext><!--/mstyle--><mi 
>R</mi>
</mrow></math> for <!--l. 978--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">

<msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->H<!--/span--></mi></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msub 
></math>-almost
every <!--l. 978--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>y</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>&#x03C0;</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></math>.
We claim that </p><table class="equation"><tr><td>
<!--l. 983--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
 <mfrac><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow>
<mrow 
><mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>t</mi><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo>  <mfrac><mrow 
><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mrow> 
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mfenced> </mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mfenced> <!--mstyle 
class="text"--><mtext >-a.e.&#x000A0;in&#x000A0;</mtext><!--/mstyle--><mi 
>R</mi><mstyle 
    class="label" id="x1-9001r34"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(34)</td></tr></table>
for <!--l. 984--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->H<!--/span--></mi></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msub 
></math>-almost
every <!--l. 984--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>y</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>&#x03C0;</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></math>.
In fact, by (<a 
href="#x1-3018r16">16<!--tex4ht:ref: sum-ali--></a>) and (<a 
href="#x1-3021r18">18<!--tex4ht:ref: delta-l--></a>) we get

<!--l. 986--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     
<mtable 
class="multline-star">
<mtr><mtd 
class="multline-star"><msub><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><msub><mrow 
><mi 
>&#x03C0;</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></mrow></msub 
> <mfrac><mrow 
><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mrow> 
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mfenced> </mrow></mfrac> <mo 
class="MathClass-punc">&#x22C5;</mo><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mfenced> <mi 
>d</mi><msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->H<!--/span--></mi></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><msub><mrow 
><mi 
>&#x03C0;</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></mrow></msub 
><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
> <mi 
>d</mi><msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->H<!--/span--></mi></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow>
</mtd></mtr><mtr><mtd 
class="multline-star"> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow> <mo 
class="MathClass-rel">=</mo>  <mfrac><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow> 
<mrow 
><mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow></mfrac><mo 
class="MathClass-punc">&#x22C5;</mo><mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><msub><mrow 
><mi 
>&#x03C0;</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></mrow></msub 
> <mfrac><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow> 
<mrow 
><mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <mo 
class="MathClass-punc">&#x22C5;</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-punc">&#x22C5;</mo><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mfenced> <mi 
>d</mi><msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->H<!--/span--></mi></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow>    </mtd></mtr></mtable>
</math>
<!--l. 994--><p class="nopar">
and (<a 
href="#x1-6017r24">24<!--tex4ht:ref: far-d--></a>) follows from (<a 
href="#x1-3015r13">13<!--tex4ht:ref: sum-Di--></a>). By the same argument it is possible to prove that
</p><table class="equation"><tr><td>
<!--l. 1000--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
 <mfrac><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>v</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow>
<mrow 
><mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>t</mi><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo>  <mfrac><mrow 
><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><msub><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mrow> 
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mfenced> </mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
></mfenced> <!--mstyle 
class="text"--><mtext >-a.e.&#x000A0;in&#x000A0;</mtext><!--/mstyle--><mi 
>R</mi><mstyle 
    class="label" id="x1-9002r35"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(35)</td></tr></table>

for <!--l. 1001--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->H<!--/span--></mi></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msub 
></math>-almost every <!--l. 1001--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>y</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>&#x03C0;</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></math>. By (<a 
href="#x1-6017r24">24<!--tex4ht:ref: far-d--></a>) and (<a 
href="#x1-6019r25">25<!--tex4ht:ref: E_SXgYy--></a>)
we get <!--l. 1003--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">        <mrow 
>
<msub><mrow 
><mo 
>lim</mo></mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-rel">&#x2192;</mo><mn>0</mn></mrow></msub 
><mfrac><mrow 
><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>t</mi><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">+</mo> <mi 
>h</mi><mfrac><mrow> <mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow>
<mrow><mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>t</mi><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <mi 
>t</mi><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
                                 <mrow 
><mi 
>h</mi></mrow></mfrac>                             <mo 
class="MathClass-rel">=</mo>  <mfrac><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>v</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow> 
<mrow 
><mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi><mo 
class="MathClass-bin">+</mo><mi 
>t</mi><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow>
</mrow></math> for <!--l. 1009--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
><!--span 
class="htf-calligraphy"-->H<!--/span--></mi></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msub 
></math>-almost every <!--l. 1009--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>y</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msub><mrow 
><mi 
>&#x03C0;</mi></mrow><mrow 
><mi 
>&#x03BD;</mi></mrow></msub 
></math>, and using again (<a 
href="#x1-3016r14">14<!--tex4ht:ref: detK1--></a>),
(<a 
href="#x1-3017r15">15<!--tex4ht:ref: detK2--></a>) we get <!--l. 1011--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">        <mrow 
>
            <msub><mrow 
><mo 
>lim</mo></mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-rel">&#x2192;</mo><mn>0</mn></mrow></msub 
><mfrac><mrow 
><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">+</mo> <mi 
>h</mi><mfrac><mrow> <mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow>
<mrow><mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
                       <mrow 
><mi 
>h</mi></mrow></mfrac>                     <mo 
class="MathClass-rel">=</mo>  <mfrac><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>v</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow> 
<mrow 
><mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow>
</mrow></math>
<!--l. 1017--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">     <mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></math>-a.e.
in <!--l. 1017--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
></math>.
   Since the function <!--l. 1019--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced><mo 
class="MathClass-bin">/</mo> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></math> is
strictly positive <!--l. 1020--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></math>-almost
everywhere, we obtain also

<!--l. 1022--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     
<mtable 
class="multline-star">
<mtr><mtd 
class="multline-star"><msub><mrow 
><mo 
>lim</mo></mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-rel">&#x2192;</mo><mn>0</mn></mrow></msub 
><mfrac><mrow 
><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">+</mo> <mi 
>h</mi><mfrac><mrow> <mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow>
 <mrow><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac>   <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mfrac><mrow> <mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow>
<mrow><mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
                               <mrow 
><mi 
>h</mi></mrow></mfrac>
</mtd></mtr><mtr><mtd 
class="multline-star"> <mo 
class="MathClass-rel">=</mo> <mfrac><mrow 
> <mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow> 
 <mrow 
><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac>   <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mfrac><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>v</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow>
<mrow 
><mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow>                                                                  </mtd></mtr></mtable>
</math>
<!--l. 1029--><p class="nopar">
<!--l. 1030--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">     <mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></math>-almost
everywhere in <!--l. 1030--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
></math>.
</p><!--l. 1032--><p class="indent">   Finally, since
<!--tex4ht:inline--></p><!--l. 1044--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
display="block"><mtable 
class="align-star">
     <mtr><mtd 
class="align-odd"></mtd>     <mtd 
class="align-even"><mfrac><mrow 
> <mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow>
 <mrow 
><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac>    <mfrac><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow> 
<mrow 
><mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow></mfrac> <mo 
class="MathClass-rel">=</mo> <mfrac><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow> 
 <mrow 
><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac>   <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="&#x3008;"  close="&#x3009;" > <mfrac><mrow 
><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mrow>
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mfenced><mspace width="2em" class="qquad"/> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><!--mstyle 
class="text"--><mtext >-a.e.&#x000A0;in&#x000A0;</mtext><!--/mstyle--><msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
> </mtd>
     </mtr><mtr><mtd 
class="align-odd"></mtd>     <mtd 
class="align-even"><mfrac><mrow 
> <mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow>
 <mrow 
><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac>    <mfrac><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>v</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow> 
<mrow 
><mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></mrow></mfrac> <mo 
class="MathClass-rel">=</mo> <mfrac><mrow 
><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>v</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mrow> 
 <mrow 
><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac>  <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="&#x3008;"  close="&#x3009;" > <mfrac><mrow 
><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>v</mi></mrow>
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mfenced><mspace width="2em" class="qquad"/> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced><!--mstyle 
class="text"--><mtext >-a.e.&#x000A0;in&#x000A0;</mtext><!--/mstyle--><msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
> </mtd>
   </mtr></mtable></math>
and since both sides of  (<a 
href="#x1-8002r33">33<!--tex4ht:ref: alimo--></a>) are zero <!--l. 1046--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></math>-almost everywhere on <!--l. 1047--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mfenced separators="" 
open="|"  close="|" ><mrow><mo 
class="MathClass-open">&#x2329;</mo><mrow><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mrow><mo 
class="MathClass-close">&#x232A;</mo></mrow></mfenced></math>-negligible sets, we

conclude that <!--l. 1048--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">       <mrow 
>
             <msub><mrow 
><mo 
>lim</mo></mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-rel">&#x2192;</mo><mn>0</mn></mrow></msub 
><mfrac><mrow 
><mi 
>f</mi> <mfenced separators="" 
open="("  close=")" ><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">+</mo> <mi 
>h</mi> <mfenced separators="" 
open="&#x3008;"  close="&#x3009;" ><mfrac><mrow> <mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi></mrow>
<mrow><mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mfenced></mfenced> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow> 
                      <mrow 
><mi 
>h</mi></mrow></mfrac>                    <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="&#x3008;"  close="&#x3009;" > <mfrac><mrow 
><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>v</mi></mrow>
<mrow 
> <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo>&#x02DC;</mo></mover><mi 
>u</mi></mfenced></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo><mi 
>&#x03BD;</mi></mfenced><mo 
class="MathClass-punc">,</mo>
</mrow></math>
<!--l. 1054--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">     <mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></math>-a.e.
in <!--l. 1054--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
></math>.
Since <!--l. 1055--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>&#x03BD;</mi></math>
is arbitrary, by Remarks <a 
href="#x1-7006r2">7.2<!--tex4ht:ref: r:dif--></a> and&#x00A0;<a 
href="#x1-7007r3">7.3<!--tex4ht:ref: r:dif0--></a> the restriction of <!--l. 1056--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>f</mi></math> to the affine space <!--l. 1057--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msubsup><mrow 
><mi 
>T</mi></mrow><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>u</mi></mrow></msubsup 
></math> is differentiable
at <!--l. 1057--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>&#x0169;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math> for <!--l. 1057--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mfenced separators="" 
open="|"  close="|" ><mover 
accent="false"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mi 
>u</mi></mfenced></math>-almost
every <!--l. 1058--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <mi 
>x</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
></math>
and (<a 
href="#x1-7002r26">26<!--tex4ht:ref: quts--></a>) holds.                                                                                    &#x25A1;
   It follows from (<a 
href="#x1-3015r13">13<!--tex4ht:ref: sum-Di--></a>), (<a 
href="#x1-3016r14">14<!--tex4ht:ref: detK1--></a>), and (<a 
href="#x1-3017r15">15<!--tex4ht:ref: detK2--></a>) that <table class="equation"><tr><td>

<!--l. 1065--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<mi 
>D</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>I</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>I</mi></mfenced><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msup 
> <mfenced separators="" 
open="|"  close="|" ><mi 
>I</mi></mfenced><msub><mrow 
><mo 
class="MathClass-op">&#x220F;</mo>
  </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>I</mi></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><msub><mrow 
> <mo 
class="MathClass-op">&#x220F;</mo>
   </mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>I</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <msub><mrow 
><mi 
>&#x03BB;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
>det</mo><msup><mrow 
><mi 
>A</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover><mo 
class="MathClass-rel">&#x2223;</mo><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-9003r36"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(36)</td></tr></table>
Let <!--l. 1066--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></math>, <!--l. 1066--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>i</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><mi 
>n</mi></math>.
Lemma 1 leads to <table class="equation"><tr><td>
<!--l. 1071--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<mi 
>D</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x220F;</mo>
  </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msub><mrow 
><mover 
accent="true"><mrow 
><mi 
>x</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
        </mrow><mrow 
><mi 
>I</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>n</mi></mrow></msub 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>I</mi></mfenced><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msup 
> <mfenced separators="" 
open="|"  close="|" ><mi 
>I</mi></mfenced><mo 
class="MathClass-op">per</mo><!--nolimits--><msup><mrow 
><mi 
>A</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>I</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>I</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
>det</mo><msup><mrow 
><mi 
>A</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover><mo 
class="MathClass-rel">&#x2223;</mo><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-9004r37"  ></mstyle><!--endlabel-->
</math>

</td><td class="eq-no">(37)</td></tr></table>
By (<a 
href="#x1-2003r3">3<!--tex4ht:ref: H-cycles--></a>), (<a 
href="#x1-3015r13">13<!--tex4ht:ref: sum-Di--></a>), and (<a 
href="#x1-9004r37">37<!--tex4ht:ref: Dx--></a>), we have the following result: <div class="newtheorem">
   <span class="head">
<a 
  name="x1-9005r2"></a>
<span 
class="cmbx-10">Theorem 7.2.</span> </span> <table class="equation"><tr><td>
<!--l. 1078--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">      
<msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo>  <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><mn>2</mn><mi 
>n</mi></mrow></mfrac><msubsup><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
     </mrow><mrow 
><mi 
>l</mi><mo 
class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow 
><mi 
>n</mi></mrow></msubsup 
><mi 
>l</mi><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mi 
>l</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msup 
><msubsup><mrow 
><mi 
>A</mi></mrow><mrow 
>
<mi 
>l</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msubsup 
><mo 
class="MathClass-punc">,</mo><mstyle 
    class="label" id="x1-9006r38"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(38)</td></tr></table>
<span 
class="cmti-10">where </span><table class="equation"><tr><td>

<!--l. 1084--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">      
<msubsup><mrow 
><mi 
>A</mi></mrow><mrow 
><mi 
>l</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msubsup 
> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
><mo 
class="MathClass-rel">&#x2286;</mo><mi 
>n</mi></mrow></msub 
><mo 
class="MathClass-op"> per</mo><!--nolimits--><msup><mrow 
><mi 
>A</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>I</mi></mrow><mrow 
>
<mi 
>l</mi></mrow></msub 
><mo 
class="MathClass-rel">&#x2223;</mo><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
>det</mo><msup><mrow 
><mi 
>A</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover></mrow><mrow 
>
<mi 
>l</mi></mrow></msub 
><mo 
class="MathClass-rel">&#x2223;</mo><msub><mrow 
><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo> <mfenced separators="" 
open="|"  close="|" ><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></mfenced> <mo 
class="MathClass-rel">=</mo> <mi 
>l</mi><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-9007r39"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(39)</td></tr></table>
   </div>
<!--l. 1087--><p class="indent">   It is worth noting that <!--l. 1087--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msubsup><mrow 
><mi 
>A</mi></mrow><mrow 
><mi 
>l</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msubsup 
></math> of
(<a 
href="#x1-9007r39">39<!--tex4ht:ref: A-l-lambda--></a>)is similar to the coefficients <!--l. 1088--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></math>
of the characteristic polynomial of (<a 
href="#x1-3012r10">10<!--tex4ht:ref: bl-sum--></a>). It is well known in graph theory that the coefficients <!--l. 1090--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></math> can
be expressed as a sum over certain subgraphs. It is interesting to see whether <!--l. 1091--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></math>, <!--l. 1091--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>&#x03BB;</mi> <mo 
class="MathClass-rel">=</mo> <mn>0</mn></math>,
structural properties of a graph.
</p><!--l. 1094--><p class="indent">   We may call (<a 
href="#x1-9006r38">38<!--tex4ht:ref: H-param--></a>) a parametric representation of <!--l. 1094--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
></math>. In computation,
the parameter <!--l. 1095--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">       <msub><mrow 
><mi 
>&#x03BB;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></math>
plays very important roles. The choice of the parameter usually
depends on the properties of the given graph. For a complete graph <!--l. 1097--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>K</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></math>, let <!--l. 1097--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>&#x03BB;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mn>1</mn></math>, <!--l. 1097--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>i</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><mi 
>n</mi></math>. It
follows from (<a 
href="#x1-9007r39">39<!--tex4ht:ref: A-l-lambda--></a>) that </p><table class="equation"><tr><td>

<!--l. 1102--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<msubsup><mrow 
><mi 
>A</mi></mrow><mrow 
><mi 
>l</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msubsup 
> <mo 
class="MathClass-rel">=</mo>  <mfenced separators="" 
open="{"  close="" ><mtable  
equalrows="false" equalcolumns="false" class="array"><mtr><mtd 
class="array"  columnalign="left"><mi 
>n</mi><mi 
>!</mi><mo 
class="MathClass-punc">,</mo></mtd><mtd 
class="array"  columnalign="left"><!--mstyle 
class="text"--><mtext >if&#x000A0;</mtext><!--/mstyle--><mi 
>l</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn>    </mtd>
</mtr><mtr><mtd 
class="array"  columnalign="left"><mn>0</mn><mo 
class="MathClass-punc">,</mo> </mtd><mtd 
class="array"  columnalign="left"><!--mstyle 
class="text"--><mtext >otherwise</mtext><!--/mstyle--><mo 
class="MathClass-punc">.</mo></mtd></mtr> <!--@{}l@{\quad }l@{}--></mtable>                                                                          </mfenced><mstyle 
    class="label" id="x1-9008r40"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(40)</td></tr></table>
By (<a 
href="#x1-9006r38">38<!--tex4ht:ref: H-param--></a>) <table class="equation"><tr><td>
<!--l. 1106--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><mn>2</mn></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>!</mi><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-9009r41"  ></mstyle><!--endlabel-->
</math>

</td><td class="eq-no">(41)</td></tr></table>
For a complete bipartite graph <!--l. 1107--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>K</mi></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mrow></msub 
></math>,
let <!--l. 1107--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">        <msub><mrow 
><mi 
>&#x03BB;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mn>0</mn></math>, <!--l. 1107--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>i</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><mi 
>n</mi></math>. By
(<a 
href="#x1-9007r39">39<!--tex4ht:ref: A-l-lambda--></a>), <table class="equation"><tr><td>
<!--l. 1114--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo>  <mfenced separators="" 
open="{"  close="" ><mtable  
equalrows="false" equalcolumns="false" class="array"><mtr><mtd 
class="array"  columnalign="left"><mo 
class="MathClass-bin">&#x2212;</mo><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mi 
>!</mi><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
><mi 
>!</mi><msub><mrow 
><mi 
>&#x03B4;</mi></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mrow></msub 
><mo 
class="MathClass-punc">,</mo></mtd><mtd 
class="array"  columnalign="left"><!--mstyle 
class="text"--><mtext >if&#x000A0;</mtext><!--/mstyle--><mi 
>l</mi> <mo 
class="MathClass-rel">=</mo> <mn>2</mn>     </mtd>
</mtr><mtr><mtd 
class="array"  columnalign="left"><mn>0</mn><mo 
class="MathClass-punc">,</mo>             </mtd><mtd 
class="array"  columnalign="left"><!--mstyle 
class="text"--><mtext >otherwise&#x000A0;</mtext><!--/mstyle--><mo 
class="MathClass-punc">.</mo></mtd></mtr> <!--@{}l@{\quad }l@{}--></mtable>                                                            </mfenced><mstyle 
    class="label" id="x1-9010r42"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(42)</td></tr></table>
Theorem &#x00A0;<a 
href="#x1-9005r2">7.2<!--tex4ht:ref: thm-H-param--></a> leads to <table class="equation"><tr><td>

<!--l. 1119--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo>     <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mrow></mfrac><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mi 
>!</mi><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
><mi 
>!</mi><msub><mrow 
><mi 
>&#x03B4;</mi></mrow><mrow 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><msub><mrow 
><mi 
>n</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mrow></msub 
><mo 
class="MathClass-punc">.</mo><mstyle 
    class="label" id="x1-9011r43"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(43)</td></tr></table>
   Now, we consider an asymmetrical approach. Theorem <a 
href="#x1-3008r3">3.3<!--tex4ht:ref: thm-main--></a> leads to
<!--l. 1122--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     
<mtable 
class="multline">
<mtr><mtd 
class="multline"></mtd><mtd><mo 
>det</mo><mi 
>K</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo>&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
><mo 
class="MathClass-punc">;</mo><mi 
>l</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>l</mi></mrow><mo 
class="MathClass-close">)</mo></mrow>
</mtd></mtr><mtr><mtd 
class="multline"></mtd><mtd> <mo 
class="MathClass-rel">=</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>I</mi><mo 
class="MathClass-rel">&#x2286;</mo><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>l</mi></mrow><mo 
class="MathClass-close">}</mo></mrow></mrow></msub 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>I</mi></mfenced></mrow></msup 
><msub><mrow 
> <mo 
class="MathClass-op">&#x220F;</mo>
   </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>I</mi></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><msub><mrow 
> <mo 
class="MathClass-op">&#x220F;</mo>
   </mrow><mrow 
><mi 
>j</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>I</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>D</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <msub><mrow 
><mi 
>&#x03BB;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
>det</mo><msup><mrow 
><mi 
>A</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover> <mo 
class="MathClass-bin">&#x222A;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>l</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mo 
class="MathClass-rel">&#x2223;</mo><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover> <mo 
class="MathClass-bin">&#x222A;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>l</mi></mrow><mo 
class="MathClass-close">}</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">.</mo></mtd><mtd><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>4</mn><mn>4</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>   </mtd></mtr></mtable>
</math>
<!--l. 1128--><p class="nopar">
</p><!--l. 1130--><p class="indent">   By (<a 
href="#x1-2003r3">3<!--tex4ht:ref: H-cycles--></a>) and (<a 
href="#x1-3018r16">16<!--tex4ht:ref: sum-ali--></a>) we have the following asymmetrical result: </p><div class="newtheorem">
<!--l. 1132--><p class="noindent"><span class="head">
<a 
  name="x1-9012r3"></a>

<span 
class="cmbx-10">Theorem 7.3.</span> </span> </p><table class="equation"><tr><td>
<!--l. 1138--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">      
<msub><mrow 
><mi 
>H</mi></mrow><mrow 
><mi 
>c</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><mn>2</mn></mrow></mfrac><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
    </mrow><mrow 
><mi 
>I</mi><mo 
class="MathClass-rel">&#x2286;</mo><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>l</mi></mrow><mo 
class="MathClass-close">}</mo></mrow></mrow></msub 
><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mfenced separators="" 
open="|"  close="|" ><mi 
>I</mi></mfenced></mrow></msup 
><mo 
class="MathClass-op"> per</mo><!--nolimits--><msup><mrow 
><mi 
>A</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>I</mi><mo 
class="MathClass-rel">&#x2223;</mo><mi 
>I</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
>det</mo><msup><mrow 
><mi 
>A</mi></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BB;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover> <mo 
class="MathClass-bin">&#x222A;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>l</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mo 
class="MathClass-rel">&#x2223;</mo><mover accent="false" 
class="mml-overline"><mrow><mi 
>I</mi></mrow><mo 
accent="true">&#x00AF;</mo></mover> <mo 
class="MathClass-bin">&#x222A;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>l</mi></mrow><mo 
class="MathClass-close">}</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow><mstyle 
    class="label" id="x1-9013r45"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(45)</td></tr></table>
<span 
class="cmti-10">which reduces to Goulden-Jackson&#x2019;s formula when </span><!--l. 1139--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>&#x03BB;</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mn>0</mn><mo 
class="MathClass-punc">,</mo><mi 
>i</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><mi 
>n</mi></math>
<span class="cite"><span 
class="cmti-10">[</span><a 
href="#Xmami:matrixth"><span 
class="cmti-10">9</span></a><span 
class="cmti-10">]</span></span><span 
class="cmti-10">.</span>
   </div>
   <h3 class="sectionHead"><span class="titlemark">8</span> <a 
  name="x1-100008"></a>Various font features of the <span 
class="cmtt-10">amsmath </span>package</h3>
   <h4 class="subsectionHead"><span class="titlemark">8.1</span> <a 
  name="x1-110008.1"></a>Bold versions of special symbols</h4>
<!--l. 1147--><p class="noindent">In the <span 
class="cmtt-10">amsmath </span>package <span 
class="cmtt-10">&#x005C;boldsymbol </span>is used for getting individual bold math
symbols and bold Greek letters--everything in math except for letters of the Latin
alphabet, where you&#x2019;d use <span 
class="cmtt-10">&#x005C;mathbf</span>. For example,

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
A&#x005F;&#x005C;infty&#x00A0;+&#x00A0;&#x005C;pi&#x00A0;A&#x005F;0&#x00A0;&#x005C;sim
&#x00A0;<br />&#x005C;mathbf{A}&#x005F;{&#x005C;boldsymbol{&#x005C;infty}}&#x00A0;&#x005C;boldsymbol{+}
&#x00A0;<br />&#x005C;boldsymbol{&#x005C;pi}&#x00A0;&#x005C;mathbf{A}&#x005F;{&#x005C;boldsymbol{0}}
</div>
</td></tr></table>
<!--l. 1156--><p class="indent">   looks like this: <!--l. 1157--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">       <mrow 
>
                                   <msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mi 
>&#x221E;</mi></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <mi 
>&#x03C0;</mi><msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
> <mo 
class="MathClass-rel">&#x223C;</mo> <msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mi 
>&#x221E;</mi></mrow></msub 
><mo 
class="MathClass-bin"> +</mo> <mi 
>&#x03C0;</mi><msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
>
</mrow></math>
</p>
   <h4 class="subsectionHead"><span class="titlemark">8.2</span> <a 
  name="x1-120008.2"></a>&#x201C;Poor man&#x2019;s bold&#x201D;</h4> If a bold version of a particular symbol doesn&#x2019;t
exist in the available fonts, then <span 
class="cmtt-10">&#x005C;boldsymbol </span>can&#x2019;t be used to make that
symbol bold. At the present time, this means that <span 
class="cmtt-10">&#x005C;boldsymbol </span>can&#x2019;t be
used with symbols from the <span 
class="cmtt-10">msam </span>and <span 
class="cmtt-10">msbm </span>fonts, among others. In some
cases, poor man&#x2019;s bold (<span 
class="cmtt-10">&#x005C;pmb</span>) can be used instead of <span 
class="cmtt-10">&#x005C;boldsymbol</span>: <!--l. 1172--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
>
                              <mfrac><mrow 
><mi 
>&#x2202;</mi><mi 
>x</mi></mrow>
<mrow 
><mi 
>&#x2202;</mi><mi 
>y</mi></mrow></mfrac><mfenced separators="" 
open="|"  close="" ></mfenced><mfenced separators="" 
open="|"  close="" ></mfenced><mfenced separators="" 
open="|"  close="" ></mfenced><mfrac><mrow 
><mi 
>&#x2202;</mi><mi 
>y</mi></mrow>
<mrow 
><mi 
>&#x2202;</mi><mi 
>z</mi></mrow></mfrac>
</mrow></math>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;[&#x005C;frac{&#x005C;partial&#x00A0;x}{&#x005C;partial&#x00A0;y}
&#x00A0;<br />&#x005C;pmb{&#x005C;bigg&#x005C;vert}
&#x00A0;<br />&#x005C;frac{&#x005C;partial&#x00A0;y}{&#x005C;partial&#x00A0;z}&#x005C;]
</div>
</td></tr></table>
<!--l. 1180--><p class="indent">   So-called &#x201C;large operator&#x201D; symbols such as <!--l. 1180--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mo 
class="MathClass-op">&#x2211;</mo>
  </math> and <!--l. 1180--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mo 
class="MathClass-op">&#x220F;</mo>
  </math>
require an additional command, <span 
class="cmtt-10">&#x005C;mathop</span>, to produce proper spacing
and limits when <span 
class="cmtt-10">&#x005C;pmb </span>is used. For further details see <span 
class="cmti-10">The </span><span class="TEX"><span 
class="cmti-10">T</span><span 
class="E"><span 
class="cmti-10">E</span></span><span 
class="cmti-10">X</span></span><span 
class="cmti-10">book</span>. <!--l. 1184--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
><msub><mrow 
>
<mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
>
<mtable class="subarray-c" rowspacing="0" columnalign="center"><mtr><mtd><mi 
>i</mi><mo 
class="MathClass-rel">&#x003C;</mo><mi 
>B</mi>
</mtd></mtr><mtr><mtd>
</mtd></mtr><mtr><mtd>
</mtd></mtr><mtr><mtd><!--mstyle 
class="text"--><mtext ></mtext><!--mstyle 
class="math"--><mi 
>i</mi><!--/mstyle--><mtext >&#x000A0;odd</mtext><!--/mstyle--> </mtd></mtr>                                                                                                 </mtable></mrow></msub 
><msub><mrow 
><mo 
class="MathClass-op">&#x220F;</mo>
                                                                        </mrow><mrow 
><mi 
>&#x03BA;</mi></mrow></msub 
><mi 
>&#x03BA;</mi><mi 
>F</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>r</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/><msub><mrow 
><mo 
class="MathClass-op">&#x2211;
&#x2211;
&#x2211;</mo>
   </mrow><mrow 
>
<mtable class="subarray-c" rowspacing="0" columnalign="center"><mtr><mtd><mi 
>i</mi><mo 
class="MathClass-rel">&#x003C;</mo><mi 
>B</mi>
</mtd></mtr><mtr><mtd>
</mtd></mtr><mtr><mtd>
</mtd></mtr><mtr><mtd><!--mstyle 
class="text"--><mtext ></mtext><!--mstyle 
class="math"--><mi 
>i</mi><!--/mstyle--><mtext >&#x000A0;odd</mtext><!--/mstyle--> </mtd></mtr>                                                                                                 </mtable></mrow></msub 
><msub><mrow 
><mo 
class="MathClass-op">&#x220F;
&#x220F;
&#x220F;</mo>
  </mrow><mrow 
><mi 
>&#x03BA;</mi></mrow></msub 
><mi 
>&#x03BA;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>r</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow>
</mrow></math>

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;[&#x005C;sum&#x005F;{&#x005C;substack{i&#x003C;B&#x005C;&#x005C;&#x005C;text{$i$&#x00A0;odd}}}
&#x00A0;<br />&#x005C;prod&#x005F;&#x005C;kappa&#x00A0;&#x005C;kappa&#x00A0;F(r&#x005F;i)&#x005C;qquad
&#x00A0;<br />&#x005C;mathop{&#x005C;pmb{&#x005C;sum}}&#x005F;{&#x005C;substack{i&#x003C;B&#x005C;&#x005C;&#x005C;text{$i$&#x00A0;odd}}}
&#x00A0;<br />&#x005C;mathop{&#x005C;pmb{&#x005C;prod}}&#x005F;&#x005C;kappa&#x00A0;&#x005C;kappa(r&#x005F;i)
&#x00A0;<br />&#x005C;]
</div>
</td></tr></table>
   <h3 class="sectionHead"><span class="titlemark">9</span> <a 
  name="x1-130009"></a>Compound symbols and other features</h3>
   <h4 class="subsectionHead"><span class="titlemark">9.1</span> <a 
  name="x1-140009.1"></a>Multiple integral signs</h4>
<!--l. 1201--><p class="noindent"><span 
class="cmtt-10">&#x005C;iint</span>, <span 
class="cmtt-10">&#x005C;iiint</span>, and <span 
class="cmtt-10">&#x005C;iiiint </span>give multiple integral signs with the spacing between
them nicely adjusted, in both text and display style. <span 
class="cmtt-10">&#x005C;idotsint </span>gives two integral
signs with dots between them.
</p><!--l. 1205--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     
<mtable 
class="gather">
<mtr> 
<mtd><munder class="msub"><mrow 
><mo 
class="MathClass-op">&#x222C;</mo></mrow><mrow 
><mi 
>A</mi></mrow></munder 
><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>x</mi><mi 
>d</mi><mi 
>y</mi><mspace width="2em" class="qquad"/><munder class="msub"><mrow 
><mo 
class="MathClass-op">&#x222D;</mo></mrow><mrow 
><mi 
>A</mi></mrow></munder 
><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi><mo 
class="MathClass-punc">,</mo><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>x</mi><mi 
>d</mi><mi 
>y</mi> <mi 
>d</mi><mi 
>z</mi></mtd>              
<mtd><mstyle 
    class="label" id="x1-14001r46"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>4</mn><mn>6</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd>
</mtr><mtr> 
<mtd><munder class="msub"><mrow 
><mo 
class="MathClass-op">&#x222C;<!--nolimits-->&#x222C;<!--nolimits--></mo></mrow><mrow 
><mi 
>A</mi></mrow></munder 
><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>w</mi><mo 
class="MathClass-punc">,</mo><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi><mo 
class="MathClass-punc">,</mo><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>w</mi><mi 
>d</mi><mi 
>x</mi><mi 
>d</mi><mi 
>y</mi> <mi 
>d</mi><mi 
>z</mi><mspace width="2em" class="qquad"/><munder class="msub"><mrow 
><mo 
class="MathClass-op">&#x222B;
  &#x22C5;&#x22C5;&#x22C5;&#x222B;</mo>
     </mrow><mrow 
><mi 
>A</mi></mrow></munder 
><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>x</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>k</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd> 
<mtd><mstyle 
    class="label" id="x1-14002r47"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>4</mn><mn>7</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd>                                      </mtr></mtable>
</math>
<!--l. 1210--><p class="nopar">
</p>

   <h4 class="subsectionHead"><span class="titlemark">9.2</span> <a 
  name="x1-150009.2"></a>Over and under arrows</h4>
<!--l. 1214--><p class="noindent">Some extra over and under arrow operations are provided in the <span 
class="cmtt-10">amsmath </span>package.
(Basic <span class="LATEX">L<span class="A">A</span><span class="TEX">T<span 
class="E">E</span>X</span></span>&#x00A0;provides <span 
class="cmtt-10">&#x005C;overrightarrow </span>and <span 
class="cmtt-10">&#x005C;overleftarrow</span>).
<!--tex4ht:inline--></p><!--l. 1224--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
display="block"><mtable 
class="align-star">
                        <mtr><mtd 
class="align-odd"><mover><mrow 
><msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mi 
>&#x03B4;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>t</mi></mrow></msub 
><mi 
>h</mi></mrow><mrow> <mo 
class="MathClass-rel">&#x2192;</mo></mrow></mover></mtd>                        <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <munder><mrow 
><msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mi 
>&#x03B4;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>t</mi></mrow></msub 
><mi 
>h</mi></mrow><mrow 
> <mi 
>&#x2192;</mi></mrow></munder></mtd>
                        </mtr><mtr><mtd 
class="align-odd"><mover><mrow 
><msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mi 
>&#x03B4;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>t</mi></mrow></msub 
><mi 
>h</mi></mrow><mrow> <mo 
class="MathClass-rel">&#x2190;</mo></mrow></mover></mtd>                        <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <munder><mrow 
><msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mi 
>&#x03B4;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>t</mi></mrow></msub 
><mi 
>h</mi></mrow><mrow 
><mi 
>&#x2190;</mi> </mrow> </munder></mtd>
                        </mtr><mtr><mtd 
class="align-odd"><mover><mrow 
><msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mi 
>&#x03B4;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>t</mi></mrow></msub 
><mi 
>h</mi></mrow><mrow> <mo 
class="MathClass-rel">&#x2194;</mo></mrow></mover></mtd>                        <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <munder><mrow 
><msub><mrow 
><mi 
>&#x03C8;</mi></mrow><mrow 
><mi 
>&#x03B4;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>t</mi></mrow></msub 
><mi 
>h</mi></mrow><mrow 
><mi 
>&#x2194;</mi> </mrow> </munder></mtd>
   </mtr></mtable></math>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{align*}
&#x00A0;<br />&#x005C;overrightarrow{&#x005C;psi&#x005F;&#x005C;delta(t)&#x00A0;E&#x005F;t&#x00A0;h}&#x0026;
&#x00A0;<br />=&#x005C;underrightarrow{&#x005C;psi&#x005F;&#x005C;delta(t)&#x00A0;E&#x005F;t&#x00A0;h}&#x005C;&#x005C;
&#x00A0;<br />&#x005C;overleftarrow{&#x005C;psi&#x005F;&#x005C;delta(t)&#x00A0;E&#x005F;t&#x00A0;h}&#x0026;
&#x00A0;<br />=&#x005C;underleftarrow{&#x005C;psi&#x005F;&#x005C;delta(t)&#x00A0;E&#x005F;t&#x00A0;h}&#x005C;&#x005C;
&#x00A0;<br />&#x005C;overleftrightarrow{&#x005C;psi&#x005F;&#x005C;delta(t)&#x00A0;E&#x005F;t&#x00A0;h}&#x0026;
&#x00A0;<br />=&#x005C;underleftrightarrow{&#x005C;psi&#x005F;&#x005C;delta(t)&#x00A0;E&#x005F;t&#x00A0;h}
&#x00A0;<br />&#x005C;end{align*}
</div>
</td></tr></table>
<!--l. 1235--><p class="indent">   These all scale properly in subscript sizes: <!--l. 1236--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
><msub><mrow 
>
                            <mo 
class="MathClass-op">&#x222B;</mo>
                              </mrow><mrow 
><mover><mrow 
><mi 
>A</mi><mi 
>B</mi></mrow><mrow><mo 
class="MathClass-rel">&#x2192;</mo></mrow></mover></mrow></msub 
><mi 
>a</mi><mi 
>x</mi><mi 
>d</mi><mi 
>x</mi>
</mrow></math>

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;[&#x005C;int&#x005F;{&#x005C;overrightarrow{AB}}&#x00A0;ax&#x005C;,dx&#x005C;]
</div>
</td></tr></table>
   <h4 class="subsectionHead"><span class="titlemark">9.3</span> <a 
  name="x1-160009.3"></a>Dots</h4>
<!--l. 1243--><p class="noindent">Normally you need only type <span 
class="cmtt-10">&#x005C;dots </span>for ellipsis dots in a math formula. The main
exception is when the dots fall at the end of the formula; then you need to specify
one of <span 
class="cmtt-10">&#x005C;dotsc </span>(series dots, after a comma), <span 
class="cmtt-10">&#x005C;dotsb </span>(binary dots, for binary relations
or operators), <span 
class="cmtt-10">&#x005C;dotsm </span>(multiplication dots), or <span 
class="cmtt-10">&#x005C;dotsi </span>(dots after an integral). For
example, the input

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
Then&#x00A0;we&#x00A0;have&#x00A0;the&#x00A0;series&#x00A0;$A&#x005F;1,A&#x005F;2,&#x005C;dotsc$,
&#x00A0;<br />the&#x00A0;regional&#x00A0;sum&#x00A0;$A&#x005F;1+A&#x005F;2+&#x005C;dotsb$,
&#x00A0;<br />the&#x00A0;orthogonal&#x00A0;product&#x00A0;$A&#x005F;1A&#x005F;2&#x005C;dotsm$,
&#x00A0;<br />and&#x00A0;the&#x00A0;infinite&#x00A0;integral
&#x00A0;<br />&#x005C;[&#x005C;int&#x005F;{A&#x005F;1}&#x005C;int&#x005F;{A&#x005F;2}&#x005C;dotsi&#x005C;].
</div>
</td></tr></table>
<!--l. 1257--><p class="indent">   produces
     </p><!--l. 1259--><p class="indent">   Then we have the series <!--l. 1259--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-op">&#x2026;</mo></math>,
     the regional sum <!--l. 1260--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
> <mo 
class="MathClass-bin">+</mo> <mo 
class="MathClass-rel">&#x22EF;</mo></math>,
     the orthogonal product <!--l. 1261--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
><mo 
class="MathClass-rel">&#x22EF;</mo> </math>,
     and the infinite integral <!--l. 1263--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">      <mrow 
><msub><mrow 
>
                                             <mo 
class="MathClass-op">&#x222B;</mo>
                       </mrow><mrow 
><msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
></mrow></msub 
><msub><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
                           </mrow><mrow 
><msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mrow></msub 
> <mo 
class="MathClass-rel">&#x22EF;</mo>
</mrow></math>
</p>
   <h4 class="subsectionHead"><span class="titlemark">9.4</span> <a 
  name="x1-170009.4"></a>Accents in math</h4>

<!--l. 1268--><p class="noindent">Double accents: <!--l. 1269--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">       <mrow 
>
                                  <mover 
accent="true"><mrow 
><mi 
>&#x0124;</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover><mover 
accent="true"><mrow 
><mi 
>&#x010C;</mi></mrow><mo 
class="MathClass-op">&#x030C;</mo></mover><mover 
accent="true"><mrow 
><mover 
accent="true"><mrow 
><mi 
>T</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover><mover 
accent="true"><mrow 
><mi 
>&#x00C1;</mi></mrow><mo 
class="MathClass-op">&#x0301;</mo></mover><mover 
accent="true"><mrow 
><mover 
accent="true"><mrow 
><mi 
>G</mi></mrow><mo 
class="MathClass-op">&#x0300;</mo></mover></mrow><mo 
class="MathClass-op">&#x0300;</mo></mover><mover 
accent="true"><mrow 
><mover 
accent="true"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x002E;</mo></mover></mrow><mo 
class="MathClass-op">&#x002E;</mo></mover><mover 
accent="true"><mrow 
><mover 
accent="true"><mrow 
><mi 
>D</mi></mrow><mo 
class="MathClass-op">&#x0308;</mo></mover></mrow><mo 
class="MathClass-op">&#x0308;</mo></mover><mover 
accent="true"><mrow 
><mover 
accent="true"><mrow 
><mi 
>B</mi></mrow><mo 
class="MathClass-op">&#x02D8;</mo></mover></mrow><mo 
class="MathClass-op">&#x02D8;</mo></mover><mover 
accent="true"><mrow 
><mover 
accent="true"><mrow 
><mi 
>B</mi></mrow><mo 
class="MathClass-op">&#x0304;</mo></mover></mrow><mo 
class="MathClass-op">&#x0304;</mo></mover><mover 
accent="true"><mrow 
><mover 
accent="true"><mrow 
><mi 
>V</mi> </mrow><mo 
class="MathClass-op">&#x2192;</mo></mover></mrow><mo 
class="MathClass-op">&#x2192;</mo></mover>
</mrow></math>

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;[&#x005C;Hat{&#x005C;Hat{H}}&#x005C;quad&#x005C;Check{&#x005C;Check{C}}&#x005C;quad
&#x00A0;<br />&#x005C;Tilde{&#x005C;Tilde{T}}&#x005C;quad&#x005C;Acute{&#x005C;Acute{A}}&#x005C;quad
&#x00A0;<br />&#x005C;Grave{&#x005C;Grave{G}}&#x005C;quad&#x005C;Dot{&#x005C;Dot{D}}&#x005C;quad
&#x00A0;<br />&#x005C;Ddot{&#x005C;Ddot{D}}&#x005C;quad&#x005C;Breve{&#x005C;Breve{B}}&#x005C;quad
&#x00A0;<br />&#x005C;Bar{&#x005C;Bar{B}}&#x005C;quad&#x005C;Vec{&#x005C;Vec{V}}&#x005C;]
</div>
</td></tr></table>
<!--l. 1281--><p class="indent">   This double accent operation is complicated and tends to slow down the
processing of a <span class="LATEX">L<span class="A">A</span><span class="TEX">T<span 
class="E">E</span>X</span></span>&#x00A0;file.
</p>
   <h4 class="subsectionHead"><span class="titlemark">9.5</span> <a 
  name="x1-180009.5"></a>Dot accents</h4> <span 
class="cmtt-10">&#x005C;dddot </span>and <span 
class="cmtt-10">&#x005C;ddddot </span>are available to produce triple and quadruple
dot accents in addition to the <span 
class="cmtt-10">&#x005C;dot </span>and <span 
class="cmtt-10">&#x005C;ddot </span>accents already available in <span class="LATEX">L<span class="A">A</span><span class="TEX">T<span 
class="E">E</span>X</span></span>: <!--l. 1290--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
>
                               <mover 
accent="true"><mrow 
><mi 
>Q</mi></mrow><mi 
>...</mi></mover><mspace width="2em" class="qquad"/><mover 
accent="true"><mrow 
><mi 
>R</mi></mrow><mi 
>....</mi></mover>
</mrow></math>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;[&#x005C;dddot{Q}&#x005C;qquad&#x005C;ddddot{R}&#x005C;]
</div>
</td></tr></table>
   <h4 class="subsectionHead"><span class="titlemark">9.6</span> <a 
  name="x1-190009.6"></a>Roots</h4>
<!--l. 1297--><p class="noindent">In the <span 
class="cmtt-10">amsmath </span>package <span 
class="cmtt-10">&#x005C;leftroot </span>and <span 
class="cmtt-10">&#x005C;uproot </span>allow you to adjust the position of
the root index of a radical:

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;sqrt[&#x005C;leftroot{-2}&#x005C;uproot{2}&#x005C;beta]{k}
</div>
</td></tr></table>
<!--l. 1302--><p class="indent">   gives good positioning of the <!--l. 1302--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">      <mi 
>&#x03B2;</mi></math>:
<!--l. 1303--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     <mrow 
>
                                                  <mroot><mrow 
><mi 
>k</mi></mrow><mrow 
><mi 
>&#x03B2;</mi></mrow></mroot>
</mrow></math>
</p>
   <h4 class="subsectionHead"><span class="titlemark">9.7</span> <a 
  name="x1-200009.7"></a>Boxed formulas</h4> The command <span 
class="cmtt-10">&#x005C;boxed </span>puts a box around its argument, like
<span 
class="cmtt-10">&#x005C;fbox </span>except that the contents are in math mode:

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;boxed{W&#x005F;t-F&#x005C;subseteq&#x00A0;V(P&#x005F;i)&#x005C;subseteq&#x00A0;W&#x005F;t}
</div>
</td></tr></table>
<!--l. 1310--><p class="indent">   <!--l. 1310--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     <mrow 
><msub><mrow 
>
<mi 
>W</mi></mrow><mrow 
><mi 
>t</mi></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>F</mi> <mo 
class="MathClass-rel">&#x2286;</mo> <mi 
>V</mi> <mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>P</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2286;</mo> <msub><mrow 
><mi 
>W</mi></mrow><mrow 
><mi 
>t</mi></mrow></msub 
>                                                                      <mo 
class="MathClass-punc">.</mo>
</mrow></math>
</p>
   <h4 class="subsectionHead"><span class="titlemark">9.8</span> <a 
  name="x1-210009.8"></a>Extensible arrows</h4> <span 
class="cmtt-10">&#x005C;xleftarrow </span>and <span 
class="cmtt-10">&#x005C;xrightarrow </span>produce arrows that extend
automatically to accommodate unusually wide subscripts or superscripts. The text of the
subscript or superscript are given as an optional resp. mandatory argument: Example: <!--l. 1318--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
>
                  <mn>0</mn><munderover><mrow 
><mi 
>&#x03B6;</mi></mrow> <mo 
class="MathClass-rel">&#x2190;</mo><mrow 
><mi 
>&#x03B1;</mi></mrow></munderover><mi 
>F</mi> <mo 
class="MathClass-bin">&#x00D7;</mo><mo 
class="MathClass-bin">&#x25B3;</mo><mrow><mo 
class="MathClass-open">[</mo><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>1</mn></mrow><mo 
class="MathClass-close">]</mo></mrow><munderover><mrow 
></mrow> <mo 
class="MathClass-rel">&#x2192;</mo><mrow 
><msub><mrow 
><mi 
>&#x2202;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mi 
>&#x03B1;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>b</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></munderover><msup><mrow 
><mi 
>E</mi></mrow><mrow 
><msub><mrow 
><mi 
>&#x2202;</mi></mrow><mrow 
><mn>0</mn></mrow></msub 
><mi 
>b</mi></mrow></msup 
>
</mrow></math>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;[0&#x00A0;&#x005C;xleftarrow[&#x005C;zeta]{&#x005C;alpha}&#x00A0;F&#x005C;times&#x005C;triangle[n-1]
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;xrightarrow{&#x005C;partial&#x005F;0&#x005C;alpha(b)}&#x00A0;E&#x02C6;{&#x005C;partial&#x005F;0b}&#x005C;]
</div>
</td></tr></table>
   <h4 class="subsectionHead"><span class="titlemark">9.9</span> <a 
  name="x1-220009.9"></a><span 
class="cmtt-10">&#x005C;overset</span>, <span 
class="cmtt-10">&#x005C;underset</span>, and <span 
class="cmtt-10">&#x005C;sideset</span></h4> Examples: <!--l. 1327--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
>
                            <mover><mrow 
><mo 
class="MathClass-bin">&#x2217;</mo></mrow><mrow 
><mi 
>X</mi></mrow></mover><mspace width="2em" class="qquad"/><munder><mrow 
> <mo 
class="MathClass-bin">&#x2217;</mo></mrow><mrow 
><mi 
>X</mi></mrow></munder><mspace width="2em" class="qquad"/><mover><mrow 
><mi 
>a</mi></mrow><mrow 
><munder><mrow 
><mi 
>b</mi></mrow><mrow 
><mi 
>X</mi></mrow></munder></mrow></mover>
</mrow></math>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;[&#x005C;overset{*}{X}&#x005C;qquad&#x005C;underset{*}{X}&#x005C;qquad
&#x00A0;<br />&#x005C;overset{a}{&#x005C;underset{b}{X}}&#x005C;]
</div>
</td></tr></table>
<!--l. 1334--><p class="indent">   The command <span 
class="cmtt-10">&#x005C;sideset </span>is for a rather special purpose: putting symbols at
the subscript and superscript corners of a large operator symbol such as <!--l. 1336--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mo 
class="MathClass-op">&#x2211;</mo>
  </math> or <!--l. 1336--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mo 
class="MathClass-op">&#x220F;</mo>
  </math>,
without affecting the placement of limits. Examples: <!--l. 1339--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
><msub><mrow 
><msubsup><mrow 
>
                       </mrow><mrow 
><mo 
class="MathClass-bin">&#x2217;</mo></mrow><mrow 
><mo 
class="MathClass-bin">&#x2217;</mo></mrow></msubsup 
><msubsup><mrow 
><mo 
class="MathClass-op">&#x220F;</mo>
  </mrow><mrow 
><mo 
class="MathClass-bin">&#x2217;</mo></mrow><mrow 
><mo 
class="MathClass-bin">&#x2217;</mo></mrow></msubsup 
></mrow><mrow 
>
<mi 
>k</mi></mrow></msub 
><mspace width="2em" class="qquad"/><msub><mrow 
><msup><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
                                 </mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mrow 
>
<mn>0</mn><mo 
class="MathClass-rel">&#x2264;</mo><mi 
>i</mi><mo 
class="MathClass-rel">&#x2264;</mo><mi 
>m</mi></mrow></msub 
><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mi 
>&#x03B2;</mi><mi 
>x</mi>
</mrow></math>

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;[&#x005C;sideset{&#x005F;*&#x02C6;*}{&#x005F;*&#x02C6;*}&#x005C;prod&#x005F;k&#x005C;qquad
&#x00A0;<br />&#x005C;sideset{}{&#x2019;}&#x005C;sum&#x005F;{0&#x005C;le&#x00A0;i&#x005C;le&#x00A0;m}&#x00A0;E&#x005F;i&#x005C;beta&#x00A0;x
&#x00A0;<br />&#x005C;]
</div>
</td></tr></table>
   <h4 class="subsectionHead"><span class="titlemark">9.10</span> <a 
  name="x1-230009.10"></a>The <span 
class="cmtt-10">&#x005C;text </span>command</h4> The main use of the command <span 
class="cmtt-10">&#x005C;text </span>is for words or phrases in a
display: <!--l. 1351--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">        <mrow 
>
                              <mi 
>y</mi> <mo 
class="MathClass-rel">=</mo> <msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
><!--mstyle 
class="text"--><mtext >if&#x000A0;and&#x000A0;only&#x000A0;if</mtext><!--/mstyle--><msubsup><mrow 
><mi 
>y</mi></mrow><mrow 
>
<mi 
>k</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msubsup 
> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>&#x03B4;</mi></mrow><mrow 
>
<mi 
>k</mi></mrow></msub 
><msub><mrow 
><mi 
>y</mi></mrow><mrow 
><mi 
>&#x03C4;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>k</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
>
</mrow></math>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;[&#x005C;mathbf{y}=&#x005C;mathbf{y}&#x2019;&#x005C;quad&#x005C;text{if&#x00A0;and&#x00A0;only&#x00A0;if}&#x005C;quad
&#x00A0;<br />y&#x2019;&#x005F;k=&#x005C;delta&#x005F;k&#x00A0;y&#x005F;{&#x005C;tau(k)}&#x005C;]
</div>
</td></tr></table>
   <h4 class="subsectionHead"><span class="titlemark">9.11</span> <a 
  name="x1-240009.11"></a>Operator names</h4> The more common math functions such as <!--l. 1359--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mo 
>log</mo><!--nolimits--></math>, <!--l. 1359--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mo 
>sin</mo><!--nolimits--></math>, and <!--l. 1359--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mo 
>lim</mo></math> have
predefined control sequences: <span class="obeylines-h"><span 
class="cmtt-10">&#x005C;log</span></span>, <span class="obeylines-h"><span 
class="cmtt-10">&#x005C;sin</span></span>, <span class="obeylines-h"><span 
class="cmtt-10">&#x005C;lim</span></span>. The <span 
class="cmtt-10">amsmath </span>package provides
<span 
class="cmtt-10">&#x005C;DeclareMathOperator </span>and <span 
class="cmtt-10">&#x005C;DeclareMathOperator* </span>for producing new
function names that will have the same typographical treatment. Examples: <!--l. 1367--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
><msub><mrow 
>
                       <mfenced separators="" 
open="|"  close="|" ><mi 
>f</mi></mfenced></mrow><mrow 
><mi 
>&#x221E;</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo><msub><mrow 
><mo 
class="MathClass-op"> esssup</mo><!--nolimits--></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-rel">&#x2208;</mo><msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
></mrow></msub 
> <mfenced separators="" 
open="|"  close="|" ><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced>
</mrow></math>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;[&#x005C;norm{f}&#x005F;&#x005C;infty=
&#x00A0;<br />&#x005C;esssup&#x005F;{x&#x005C;in&#x00A0;R&#x02C6;n}&#x005C;abs{f(x)}&#x005C;]
</div>
</td></tr></table>
<!--l. 1373--><p class="indent">   <!--l. 1373--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     <mrow 
>
     <msub><mrow 
><mo 
class="MathClass-op">meas</mo><!--nolimits--></mrow><mrow 
><mn>1</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>u</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msubsup><mrow 
><mi 
>R</mi></mrow><mrow 
><mo 
class="MathClass-bin">+</mo></mrow><mrow 
><mn>1</mn></mrow></msubsup 
><mo 
class="MathClass-punc">:</mo> <msup><mrow 
><mi 
>f</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2217;</mo></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>u</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x003E;</mo> <mi 
>&#x03B1;</mi></mrow><mo 
class="MathClass-close">}</mo></mrow> <mo 
class="MathClass-rel">=</mo><msub><mrow 
><mo 
class="MathClass-op"> meas</mo><!--nolimits--></mrow><mrow 
>
<mi 
>n</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>x</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <msup><mrow 
><mi 
>R</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
><mo 
class="MathClass-punc">:</mo>  <mfenced separators="" 
open="|"  close="|" ><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced> <mo 
class="MathClass-rel">&#x2265;</mo> <mi 
>&#x03B1;</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mi 
>&#x2200;</mi><mi 
>&#x03B1;</mi> <mo 
class="MathClass-rel">&#x003E;</mo> <mn>0</mn><mo 
class="MathClass-punc">.</mo>
</mrow></math>

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;[&#x005C;meas&#x005F;1&#x005C;{u&#x005C;in&#x00A0;R&#x005F;+&#x02C6;1&#x005C;colon&#x00A0;f&#x02C6;*(u)&#x003E;&#x005C;alpha&#x005C;}
&#x00A0;<br />=&#x005C;meas&#x005F;n&#x005C;{x&#x005C;in&#x00A0;R&#x02C6;n&#x005C;colon&#x00A0;&#x005C;abs{f(x)}&#x005C;geq&#x005C;alpha&#x005C;}
&#x00A0;<br />&#x005C;quad&#x00A0;&#x005C;forall&#x005C;alpha&#x003E;0.&#x005C;]
</div>
</td></tr></table>
<!--l. 1381--><p class="indent">   <span 
class="cmtt-10">&#x005C;esssup </span>and <span 
class="cmtt-10">&#x005C;meas </span>would be defined in the document preamble as

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;DeclareMathOperator*{&#x005C;esssup}{ess&#x005C;,sup}
&#x00A0;<br />&#x005C;DeclareMathOperator{&#x005C;meas}{meas}
</div>
</td></tr></table>
<!--l. 1387--><p class="indent">   The following special operator names are predefined in the <span 
class="cmtt-10">amsmath </span>package:
<span 
class="cmtt-10">&#x005C;varlimsup</span>, <span 
class="cmtt-10">&#x005C;varliminf</span>, <span 
class="cmtt-10">&#x005C;varinjlim</span>, and <span 
class="cmtt-10">&#x005C;varprojlim</span>. Here&#x2019;s what they look like
in use:
<!--tex4ht:inline--></p><!--l. 1397--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
display="block"><mtable 
class="align">
                     <mtr><mtd 
class="align-odd"></mtd>                     <mtd 
class="align-even"><msub><mrow 
><mover accent="false" 
class="mml-overline"><mrow><mo 
>lim</mo></mrow><mo 
accent="true">&#x00AF;</mo></mover></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-rel">&#x2192;</mo><mi 
>&#x221E;</mi></mrow></msub 
><mi 
><!--span 
class="htf-calligraphy"-->Q<!--/span--></mi><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msup><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>#</mi></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2264;</mo> <mn>0</mn> </mtd>                     <mtd 
class="align-label">                            <mstyle 
    class="label" id="x1-24001r48"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>4</mn><mn>8</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
                     </mtd></mtr><mtr><mtd 
class="align-odd"></mtd>                     <mtd 
class="align-even"><msub><mrow 
><munder class="mml-underline"><mrow><mo 
>lim</mo></mrow><mo 
accent="true">&OverBar;</mo></munder></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-rel">&#x2192;</mo><mi 
>&#x221E;</mi></mrow></msub 
><mfenced separators="" 
open="|"  close="|" ><msub><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">+</mo><mn>1</mn></mrow></msub 
></mfenced> <mo 
class="MathClass-bin">/</mo> <mfenced separators="" 
open="|"  close="|" ><msub><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mfenced> <mo 
class="MathClass-rel">=</mo> <mn>0</mn></mtd>                         <mtd 
class="align-label">                            <mstyle 
    class="label" id="x1-24002r49"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>4</mn><mn>9</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
                     </mtd></mtr><mtr><mtd 
class="align-odd"></mtd>                     <mtd 
class="align-even"><munder><mrow 
><mo 
>lim</mo></mrow><mrow 
> <mi 
>&#x2192;</mi></mrow></munder><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msubsup><mrow 
><mi 
>m</mi></mrow><mrow 
><mi 
>i</mi></mrow><mrow 
><mi 
>&#x03BB;</mi></mrow></msubsup 
><mo 
class="MathClass-punc">&#x22C5;</mo></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mo 
class="MathClass-bin">&#x2217;</mo></mrow></msup 
><mo 
class="MathClass-rel">&#x2264;</mo> <mn>0</mn></mtd>                               <mtd 
class="align-label">                            <mstyle 
    class="label" id="x1-24003r50"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>5</mn><mn>0</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
                     </mtd></mtr><mtr><mtd 
class="align-odd"></mtd>                     <mtd 
class="align-even"><msub><mrow 
><munder><mrow 
><mo 
>lim</mo></mrow><mrow 
><mi 
>&#x2190;</mi> </mrow> </munder> </mrow><mrow 
><mi 
>p</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>S</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>A</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><msub><mrow 
><mi 
>A</mi></mrow><mrow 
><mi 
>p</mi></mrow></msub 
> <mo 
class="MathClass-rel">&#x2264;</mo> <mn>0</mn></mtd>                              <mtd 
class="align-label">                            <mstyle 
    class="label" id="x1-24004r51"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>5</mn><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
   </mtd></mtr></mtable></math>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{align}
&#x00A0;<br />&#x0026;&#x005C;varlimsup&#x005F;{n&#x005C;rightarrow&#x005C;infty}
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x005C;mathcal{Q}(u&#x005F;n,u&#x005F;n-u&#x02C6;{&#x005C;#})&#x005C;le0&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;varliminf&#x005F;{n&#x005C;rightarrow&#x005C;infty}
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;left&#x005C;lvert&#x00A0;a&#x005F;{n+1}&#x005C;right&#x005C;rvert/&#x005C;left&#x005C;lvert&#x00A0;a&#x005F;n&#x005C;right&#x005C;rvert=0&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;varinjlim&#x00A0;(m&#x005F;i&#x02C6;&#x005C;lambda&#x005C;cdot)&#x02C6;*&#x005C;le0&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;varprojlim&#x005F;{p&#x005C;in&#x00A0;S(A)}A&#x005F;p&#x005C;le0
&#x00A0;<br />&#x005C;end{align}
</div>
</td></tr></table>
   <h4 class="subsectionHead"><span class="titlemark">9.12</span> <a 
  name="x1-250009.12"></a><span 
class="cmtt-10">&#x005C;mod </span>and its relatives</h4> The commands <span 
class="cmtt-10">&#x005C;mod </span>and <span 
class="cmtt-10">&#x005C;pod </span>are variants of <span 
class="cmtt-10">&#x005C;pmod</span>
preferred by some authors; <span 
class="cmtt-10">&#x005C;mod </span>omits the parentheses, whereas <span 
class="cmtt-10">&#x005C;pod </span>omits the
&#x2018;mod&#x2019; and retains the parentheses. Examples:
<!--tex4ht:inline--><!--l. 1418--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
display="block"><mtable 
class="align">
                        <mtr><mtd 
class="align-odd"><mi 
>x</mi></mtd>                        <mtd 
class="align-even"> <mo 
class="MathClass-rel">&#x2261;</mo> <mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <mn>1</mn>   <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>m</mi><mi 
>o</mi><mi 
>d</mi> <msup><mrow 
><mi 
>m</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd>                        <mtd 
class="align-label">                                <mstyle 
    class="label" id="x1-25001r52"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>5</mn><mn>2</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
                        </mtd></mtr><mtr><mtd 
class="align-odd"><mi 
>x</mi></mtd>                        <mtd 
class="align-even"> <mo 
class="MathClass-rel">&#x2261;</mo> <mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <mn>1</mn>   <mi 
>m</mi><mi 
>o</mi><mi 
>d</mi> <msup><mrow 
><mi 
>m</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
></mtd>                         <mtd 
class="align-label">                                        <mstyle 
    class="label" id="x1-25002r53"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>5</mn><mn>3</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
                        </mtd></mtr><mtr><mtd 
class="align-odd"><mi 
>x</mi></mtd>                        <mtd 
class="align-even"> <mo 
class="MathClass-rel">&#x2261;</mo> <mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <mn>1</mn>   <mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>m</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd>                            <mtd 
class="align-label">                                <mstyle 
    class="label" id="x1-25003r54"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>5</mn><mn>4</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
   </mtd></mtr></mtable></math>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{align}
&#x00A0;<br />x&#x0026;&#x005C;equiv&#x00A0;y+1&#x005C;pmod{m&#x02C6;2}&#x005C;&#x005C;
&#x00A0;<br />x&#x0026;&#x005C;equiv&#x00A0;y+1&#x005C;mod{m&#x02C6;2}&#x005C;&#x005C;
&#x00A0;<br />x&#x0026;&#x005C;equiv&#x00A0;y+1&#x005C;pod{m&#x02C6;2}
&#x00A0;<br />&#x005C;end{align}
</div>
</td></tr></table>
   <h4 class="subsectionHead"><span class="titlemark">9.13</span> <a 
  name="x1-260009.13"></a>Fractions and related constructions</h4>
<!--l. 1430--><p class="noindent">The usual notation for binomials is similar to the fraction concept, so it has a similar
command <span 
class="cmtt-10">&#x005C;binom </span>with two arguments. Example: </p><table class="equation"><tr><td>
<!--l. 1440--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow> <msub><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>&#x03B3;</mi><mo 
class="MathClass-rel">&#x2208;</mo><msub><mrow 
><mi 
>&#x0393;</mi></mrow><mrow 
><mi 
>C</mi></mrow></msub 
></mrow></msub 
><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mi 
>&#x03B3;</mi></mrow></msub 
><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <msup><mrow 
><mn>2</mn></mrow><mrow 
><mi 
>k</mi></mrow></msup 
> <mo 
class="MathClass-bin">&#x2212;</mo><mfenced separators="" 
open="(" close=")"><mfrac linethickness="0.0pt"><mrow> <mi 
>k</mi></mrow> 
<mrow><mn>1</mn></mrow></mfrac></mfenced><msup><mrow 
><mn>2</mn></mrow><mrow 
><mi 
>k</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msup 
> <mo 
class="MathClass-bin">+</mo><mfenced separators="" 
open="(" close=")"><mfrac linethickness="0.0pt"><mrow> <mi 
>k</mi></mrow> 
<mrow><mn>2</mn></mrow></mfrac></mfenced><msup><mrow 
><mn>2</mn></mrow><mrow 
><mi 
>k</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>2</mn></mrow></msup 
>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>                    <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-bin">+</mo> <mo 
class="MathClass-rel">&#x22EF;</mo> <mo 
class="MathClass-bin">+</mo> <msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mi 
>l</mi></mrow></msup 
><mfenced separators="" 
open="(" close=")"><mfrac linethickness="0.0pt"><mrow><mi 
>k</mi></mrow> 
<mrow><mi 
>l</mi></mrow></mfrac></mfenced> <msup><mrow 
><mn>2</mn></mrow><mrow 
><mi 
>k</mi><mo 
class="MathClass-bin">&#x2212;</mo><mi 
>l</mi></mrow></msup 
> <mo 
class="MathClass-bin">+</mo> <mo 
class="MathClass-rel">&#x22EF;</mo> <mo 
class="MathClass-bin">+</mo> <msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mi 
>k</mi></mrow></msup 
>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>                <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>2</mn> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mi 
>k</mi></mrow></msup 
> <mo 
class="MathClass-rel">=</mo> <mn>1</mn>
    </mtd></mtr></mtable>                                                                                                   <mstyle 
    class="label" id="x1-26001r55"  ></mstyle><!--endlabel-->
</math>

</td><td class="eq-no">(55)</td></tr></table>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{equation}
&#x00A0;<br />&#x005C;begin{split}
&#x00A0;<br />[&#x005C;sum&#x005F;{&#x005C;gamma&#x005C;in&#x005C;Gamma&#x005F;C}&#x00A0;I&#x005F;&#x005C;gamma&#x0026;
&#x00A0;<br />=2&#x02C6;k-&#x005C;binom{k}{1}2&#x02C6;{k-1}+&#x005C;binom{k}{2}2&#x02C6;{k-2}&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;quad+&#x005C;dots+(-1)&#x02C6;l&#x005C;binom{k}{l}2&#x02C6;{k-l}
&#x00A0;<br />+&#x005C;dots+(-1)&#x02C6;k&#x005C;&#x005C;
&#x00A0;<br />&#x0026;=(2-1)&#x02C6;k=1
&#x00A0;<br />&#x005C;end{split}
&#x00A0;<br />&#x005C;end{equation}
</div>
</td></tr></table>
<!--l. 1452--><p class="indent">   There are also abbreviations

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;dfrac&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x005C;dbinom
&#x00A0;<br />&#x005C;tfrac&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x005C;tbinom
</div>
</td></tr></table>
<!--l. 1457--><p class="indent">   for the commonly needed constructions

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
{&#x005C;displaystyle&#x005C;frac&#x00A0;...&#x00A0;}&#x00A0;&#x00A0;&#x00A0;{&#x005C;displaystyle&#x005C;binom&#x00A0;...&#x00A0;}
&#x00A0;<br />{&#x005C;textstyle&#x005C;frac&#x00A0;...&#x00A0;}&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;{&#x005C;textstyle&#x005C;binom&#x00A0;...&#x00A0;}
</div>
</td></tr></table>
<!--l. 1463--><p class="indent">   The generalized fraction command <span 
class="cmtt-10">&#x005C;genfrac </span>provides full access to the six <span class="TEX">T<span 
class="E">E</span>X</span>
fraction primitives:
<!--tex4ht:inline--></p><!--l. 1475--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
display="block"><mtable 
class="align">
         <mtr><mtd 
class="align-odd"><!--mstyle 
class="text"--><mtext >&#x005C;over:&#x000A0;</mtext><!--/mstyle--></mtd>          <mtd 
class="align-even"><mfrac><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">+</mo> <mn>1</mn></mrow>
   <mrow><mn>2</mn></mrow></mfrac>   </mtd>         <mtd 
class="align-odd"><!--mstyle 
class="text"--><mtext >&#x005C;overwithdelims:&#x000A0;</mtext><!--/mstyle--></mtd>          <mtd 
class="align-even"><mfenced separators="" 
open="&#x3008;" close="&#x3009;"><mfrac><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">+</mo> <mn>1</mn></mrow>
   <mrow><mn>2</mn></mrow></mfrac></mfenced></mtd>                             <mtd 
class="align-label">         <mstyle 
    class="label" id="x1-26002r56"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>5</mn><mn>6</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
         </mtd></mtr><mtr><mtd 
class="align-odd"><!--mstyle 
class="text"--><mtext >&#x005C;atop:&#x000A0;</mtext><!--/mstyle--></mtd>          <mtd 
class="align-even"><mfrac linethickness="0.0pt"><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">+</mo> <mn>1</mn></mrow>
   <mrow><mn>2</mn></mrow></mfrac>   </mtd>         <mtd 
class="align-odd"><!--mstyle 
class="text"--><mtext >&#x005C;atopwithdelims:&#x000A0;</mtext><!--/mstyle--></mtd>          <mtd 
class="align-even"><mfenced separators="" 
open="(" close=")"><mfrac linethickness="0.0pt"><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">+</mo> <mn>1</mn></mrow>
   <mrow><mn>2</mn></mrow></mfrac></mfenced></mtd>                             <mtd 
class="align-label">         <mstyle 
    class="label" id="x1-26003r57"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>5</mn><mn>7</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
         </mtd></mtr><mtr><mtd 
class="align-odd"><!--mstyle 
class="text"--><mtext >&#x005C;above:&#x000A0;</mtext><!--/mstyle--></mtd>         <mtd 
class="align-even"><mfrac linethickness="1.0pt"><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">+</mo> <mn>1</mn></mrow>
   <mrow><mn>2</mn></mrow></mfrac>   </mtd>         <mtd 
class="align-odd"><!--mstyle 
class="text"--><mtext >&#x005C;abovewithdelims:&#x000A0;</mtext><!--/mstyle--></mtd>         <mtd 
class="align-even"><mfenced separators="" 
open="[" close="]"><mfrac linethickness="1.0pt"><mrow><mi 
>n</mi> <mo 
class="MathClass-bin">+</mo> <mn>1</mn></mrow>
   <mrow><mn>2</mn></mrow></mfrac></mfenced></mtd>                             <mtd 
class="align-label">         <mstyle 
    class="label" id="x1-26004r58"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>5</mn><mn>8</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
   </mtd></mtr></mtable></math>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;text{&#x005C;cn{over}:&#x00A0;}&#x0026;&#x005C;genfrac{}{}{}{}{n+1}{2}&#x0026;
&#x00A0;<br />&#x005C;text{&#x005C;cn{overwithdelims}:&#x00A0;}&#x0026;
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;genfrac{&#x005C;langle}{&#x005C;rangle}{}{}{n+1}{2}&#x005C;&#x005C;
&#x00A0;<br />&#x005C;text{&#x005C;cn{atop}:&#x00A0;}&#x0026;&#x005C;genfrac{}{}{0pt}{}{n+1}{2}&#x0026;
&#x00A0;<br />&#x005C;text{&#x005C;cn{atopwithdelims}:&#x00A0;}&#x0026;
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;genfrac{(}{)}{0pt}{}{n+1}{2}&#x005C;&#x005C;
&#x00A0;<br />&#x005C;text{&#x005C;cn{above}:&#x00A0;}&#x0026;&#x005C;genfrac{}{}{1pt}{}{n+1}{2}&#x0026;
&#x00A0;<br />&#x005C;text{&#x005C;cn{abovewithdelims}:&#x00A0;}&#x0026;
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;genfrac{[}{]}{1pt}{}{n+1}{2}
</div>
</td></tr></table>
   <h4 class="subsectionHead"><span class="titlemark">9.14</span> <a 
  name="x1-270009.14"></a>Continued fractions</h4> The continued fraction <table class="equation"><tr><td>
<!--l. 1497--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
                   <mfrac><mrow 
><mn>1</mn></mrow>
<mrow 
><msqrt><!--<mi 
>&#x221A;</mi>
 --><mrow><mn>2</mn></mrow></msqrt> <mo 
class="MathClass-bin">+</mo>                <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><msqrt><!--<mi 
>&#x221A;</mi>
 --><mrow><mn>2</mn></mrow></msqrt> <mo 
class="MathClass-bin">+</mo>            <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><msqrt><!--<mi 
>&#x221A;</mi>
 --><mrow><mn>2</mn></mrow></msqrt> <mo 
class="MathClass-bin">+</mo>        <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><msqrt><!--<mi 
>&#x221A;</mi>
 --><mrow><mn>2</mn></mrow></msqrt> <mo 
class="MathClass-bin">+</mo>    <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><msqrt><!--<mi 
>&#x221A;</mi>
 --><mrow><mn>2</mn></mrow></msqrt> <mo 
class="MathClass-bin">+</mo> <mo 
class="MathClass-rel">&#x22EF;</mo></mrow></mfrac></mrow></mfrac></mrow></mfrac></mrow></mfrac></mrow></mfrac><mstyle 
    class="label" id="x1-27001r59"  ></mstyle><!--endlabel-->
</math>

</td><td class="eq-no">(59)</td></tr></table>
can be obtained by typing

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;cfrac{1}{&#x005C;sqrt{2}+
&#x00A0;<br />&#x00A0;&#x005C;cfrac{1}{&#x005C;sqrt{2}+
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;cfrac{1}{&#x005C;sqrt{2}+
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;&#x005C;cfrac{1}{&#x005C;sqrt{2}+
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x005C;cfrac{1}{&#x005C;sqrt{2}+&#x005C;dotsb
&#x00A0;<br />}}}}}
</div>
</td></tr></table>
<!--l. 1507--><p class="indent">   Left or right placement of any of the numerators is accomplished by using
<span 
class="cmtt-10">&#x005C;cfrac[l] </span>or <span 
class="cmtt-10">&#x005C;cfrac[r] </span>instead of <span 
class="cmtt-10">&#x005C;cfrac</span>.
</p>
   <h4 class="subsectionHead"><span class="titlemark">9.15</span> <a 
  name="x1-280009.15"></a>Smash</h4>
<!--l. 1512--><p class="noindent">In <span 
class="cmtt-10">amsmath </span>there are optional arguments <span class="obeylines-h"><span 
class="cmtt-10">t</span></span> and <span class="obeylines-h"><span 
class="cmtt-10">b</span></span> for the plain <span class="TEX">T<span 
class="E">E</span>X</span>&#x00A0;command <span 
class="cmtt-10">&#x005C;smash</span>,
because sometimes it is advantageous to be able to &#x2018;smash&#x2019; only the top or only the
bottom of something while retaining the natural depth or height. In the formula <!--l. 1516--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>X</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><mn>1</mn><mo 
class="MathClass-bin">/</mo><msqrt><!--<mi 
>&#x221A;</mi>
 --><mrow><msub><mrow 
><mi 
>&#x03BB;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
></mrow></msqrt></mrow><mo 
class="MathClass-close">)</mo></mrow><msubsup><mrow 
><mi 
>X</mi></mrow><mrow 
><mi 
>j</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msubsup 
></math>
<span 
class="cmtt-10">&#x005C;smash</span><span class="obeylines-h"><span 
class="cmtt-10">[b]</span></span> has been used to limit the size of the radical symbol.

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
$X&#x005F;j=(1/&#x005C;sqrt{&#x005C;smash[b]{&#x005C;lambda&#x005F;j}})X&#x005F;j&#x2019;$
</div>
</td></tr></table>
<!--l. 1521--><p class="indent">   Without the use of <span 
class="cmtt-10">&#x005C;smash</span><span class="obeylines-h"><span 
class="cmtt-10">[b]</span></span> the formula would have appeared thus: <!--l. 1522--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<msub><mrow 
><mi 
>X</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><mn>1</mn><mo 
class="MathClass-bin">/</mo><msqrt><!--<mi 
>&#x2218;</mi>
 --><mrow><msub><mrow 
><mi 
>&#x03BB;</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
></mrow></msqrt></mrow><mo 
class="MathClass-close">)</mo></mrow><msubsup><mrow 
><mi 
>X</mi></mrow><mrow 
><mi 
>j</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msubsup 
></math>,
with the radical extending to encompass the depth of the subscript <!--l. 1523--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mi 
>j</mi></math>.
</p>
   <h4 class="subsectionHead"><span class="titlemark">9.16</span> <a 
  name="x1-290009.16"></a>The &#x2018;cases&#x2019; environment</h4> &#x2018;Cases&#x2019; constructions like the following can be
produced using the <span 
class="cmtt-10">cases </span>environment. <table class="equation"><tr><td>
<!--l. 1534--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<msub><mrow 
><mi 
>P</mi></mrow><mrow 
><mi 
>r</mi><mo 
class="MathClass-bin">&#x2212;</mo><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-rel">=</mo>  <mfenced separators="" 
open="{"  close="" ><mtable  
equalrows="false" equalcolumns="false" class="array"><mtr><mtd 
class="array"  columnalign="left"><mn>0</mn>                </mtd><mtd 
class="array"  columnalign="left"><!--mstyle 
class="text"--><mtext >if&#x000A0;</mtext><!--mstyle 
class="math"--><mi 
>r</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>j</mi><!--/mstyle--><mtext >&#x000A0;is&#x000A0;odd</mtext><!--/mstyle--><mo 
class="MathClass-punc">,</mo> </mtd>
</mtr><mtr><mtd 
class="array"  columnalign="left"><mi 
>r</mi><mi 
>!</mi><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>r</mi><mo 
class="MathClass-bin">&#x2212;</mo><mi 
>j</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-bin">/</mo><mn>2</mn></mrow></msup 
></mtd><mtd 
class="array"  columnalign="left"><!--mstyle 
class="text"--><mtext >if&#x000A0;</mtext><!--mstyle 
class="math"--><mi 
>r</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>j</mi><!--/mstyle--><mtext >&#x000A0;is&#x000A0;even</mtext><!--/mstyle--><mo 
class="MathClass-punc">.</mo></mtd></mtr> <!--@{}l@{\quad }l@{}--></mtable>                                                        </mfenced><mstyle 
    class="label" id="x1-29001r60"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(60)</td></tr></table>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{equation}&#x00A0;P&#x005F;{r-j}=
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;begin{cases}
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;&#x00A0;0&#x0026;&#x00A0;&#x00A0;&#x005C;text{if&#x00A0;$r-j$&#x00A0;is&#x00A0;odd},&#x005C;&#x005C;
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;&#x00A0;r!&#x005C;,(-1)&#x02C6;{(r-j)/2}&#x0026;&#x00A0;&#x00A0;&#x005C;text{if&#x00A0;$r-j$&#x00A0;is&#x00A0;even}.
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;end{cases}
&#x00A0;<br />&#x005C;end{equation}
</div>
</td></tr></table>
<!--l. 1543--><p class="indent">   Notice the use of <span 
class="cmtt-10">&#x005C;text </span>and the embedded math.
</p>
   <h4 class="subsectionHead"><span class="titlemark">9.17</span> <a 
  name="x1-300009.17"></a>Matrix</h4>
<!--l. 1547--><p class="noindent">Here are samples of the matrix environments, <span 
class="cmtt-10">&#x005C;matrix</span>, <span 
class="cmtt-10">&#x005C;pmatrix</span>, <span 
class="cmtt-10">&#x005C;bmatrix</span>,
<span 
class="cmtt-10">&#x005C;Bmatrix</span>, <span 
class="cmtt-10">&#x005C;vmatrix </span>and <span 
class="cmtt-10">&#x005C;Vmatrix</span>: </p><table class="equation"><tr><td>
<!--l. 1569--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">    
<mtable  
equalrows="false" equalcolumns="false" class="array"><mtr><mtd 
class="array"  columnalign="center"><mi 
>&#x03D1;</mi></mtd> <mtd 
class="array"  columnalign="center"><mi 
>&#x03F1;</mi></mtd>
</mtr><mtr><mtd 
class="array"  columnalign="center"><mi 
>&#x03D5;</mi></mtd><mtd 
class="array"  columnalign="center"><mi 
>&#x03D6;</mi></mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable>                                                                                   <mfenced separators="" 
open="("  close=")" ><mtable  
equalrows="false" equalcolumns="false" class="array"><mtr><mtd 
class="array"  columnalign="center"><mi 
>&#x03D1;</mi></mtd> <mtd 
class="array"  columnalign="center"><mi 
>&#x03F1;</mi></mtd>
</mtr><mtr><mtd 
class="array"  columnalign="center"><mi 
>&#x03D5;</mi></mtd><mtd 
class="array"  columnalign="center"><mi 
>&#x03D6;</mi></mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable>                                                                                   </mfenced> <mfenced separators="" 
open="["  close="]" ><mtable  
equalrows="false" equalcolumns="false" class="array"><mtr><mtd 
class="array"  columnalign="center"><mi 
>&#x03D1;</mi></mtd> <mtd 
class="array"  columnalign="center"><mi 
>&#x03F1;</mi></mtd>
</mtr><mtr><mtd 
class="array"  columnalign="center"><mi 
>&#x03D5;</mi></mtd><mtd 
class="array"  columnalign="center"><mi 
>&#x03D6;</mi></mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable>                                                                                   </mfenced> <mfenced separators="" 
open="{"  close="}" ><mtable  
equalrows="false" equalcolumns="false" class="array"><mtr><mtd 
class="array"  columnalign="center"><mi 
>&#x03D1;</mi></mtd> <mtd 
class="array"  columnalign="center"><mi 
>&#x03F1;</mi></mtd>
</mtr><mtr><mtd 
class="array"  columnalign="center"><mi 
>&#x03D5;</mi></mtd><mtd 
class="array"  columnalign="center"><mi 
>&#x03D6;</mi></mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable>                                                                                   </mfenced> <mfenced separators="" 
open="|"  close="|" ><mtable  
equalrows="false" equalcolumns="false" class="array"><mtr><mtd 
class="array"  columnalign="center"><mi 
>&#x03D1;</mi></mtd> <mtd 
class="array"  columnalign="center"><mi 
>&#x03F1;</mi></mtd>
</mtr><mtr><mtd 
class="array"  columnalign="center"><mi 
>&#x03D5;</mi></mtd><mtd 
class="array"  columnalign="center"><mi 
>&#x03D6;</mi></mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable>                                                                                   </mfenced> <mfenced separators="" 
open="|"  close="|" ><mtable  
equalrows="false" equalcolumns="false" class="array"><mtr><mtd 
class="array"  columnalign="center"><mi 
>&#x03D1;</mi></mtd> <mtd 
class="array"  columnalign="center"><mi 
>&#x03F1;</mi></mtd>
</mtr><mtr><mtd 
class="array"  columnalign="center"><mi 
>&#x03D5;</mi></mtd><mtd 
class="array"  columnalign="center"><mi 
>&#x03D6;</mi></mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable>                                                                                   </mfenced><mstyle 
    class="label" id="x1-30001r61"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(61)</td></tr></table>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{matrix}
&#x00A0;<br />&#x005C;vartheta&#x0026;&#x00A0;&#x005C;varrho&#x005C;&#x005C;&#x005C;varphi&#x0026;&#x00A0;&#x005C;varpi
&#x00A0;<br />&#x005C;end{matrix}&#x005C;quad
&#x00A0;<br />&#x005C;begin{pmatrix}
&#x00A0;<br />&#x005C;vartheta&#x0026;&#x00A0;&#x005C;varrho&#x005C;&#x005C;&#x005C;varphi&#x0026;&#x00A0;&#x005C;varpi
&#x00A0;<br />&#x005C;end{pmatrix}&#x005C;quad
&#x00A0;<br />&#x005C;begin{bmatrix}
&#x00A0;<br />&#x005C;vartheta&#x0026;&#x00A0;&#x005C;varrho&#x005C;&#x005C;&#x005C;varphi&#x0026;&#x00A0;&#x005C;varpi
&#x00A0;<br />&#x005C;end{bmatrix}&#x005C;quad
&#x00A0;<br />&#x005C;begin{Bmatrix}
&#x00A0;<br />&#x005C;vartheta&#x0026;&#x00A0;&#x005C;varrho&#x005C;&#x005C;&#x005C;varphi&#x0026;&#x00A0;&#x005C;varpi
&#x00A0;<br />&#x005C;end{Bmatrix}&#x005C;quad
&#x00A0;<br />&#x005C;begin{vmatrix}
&#x00A0;<br />&#x005C;vartheta&#x0026;&#x00A0;&#x005C;varrho&#x005C;&#x005C;&#x005C;varphi&#x0026;&#x00A0;&#x005C;varpi
&#x00A0;<br />&#x005C;end{vmatrix}&#x005C;quad
&#x00A0;<br />&#x005C;begin{Vmatrix}
&#x00A0;<br />&#x005C;vartheta&#x0026;&#x00A0;&#x005C;varrho&#x005C;&#x005C;&#x005C;varphi&#x0026;&#x00A0;&#x005C;varpi
&#x00A0;<br />&#x005C;end{Vmatrix}
</div>
</td></tr></table>
<!--l. 1592--><p class="indent">   To produce a small matrix suitable for use in text, use the <span 
class="cmtt-10">smallmatrix</span>
environment.

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{math}
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;bigl(&#x00A0;&#x005C;begin{smallmatrix}
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;a&#x0026;b&#x005C;&#x005C;&#x00A0;c&#x0026;d
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x005C;end{smallmatrix}&#x00A0;&#x005C;bigr)
&#x00A0;<br />&#x005C;end{math}
</div>
</td></tr></table>
<!--l. 1601--><p class="indent">   To show the effect of the matrix on the surrounding lines of a paragraph, we put it here: <!--l. 1603--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="inline">
<mrow><mo 
class="MathClass-open">(</mo><mrow><mtable><mtr 
class="smallmatrix"> 
<mtd><mi 
>a</mi></mtd> 
<mtd><mi 
>b</mi></mtd>
</mtr><mtr 
class="smallmatrix"> 
<mtd></mtd>
</mtr><mtr 
class="smallmatrix"> 
<mtd></mtd>
</mtr><mtr 
class="smallmatrix"> 
<mtd><mi 
>c</mi></mtd> 
<mtd><mi 
>d</mi></mtd> 
</mtr>                                                               </mtable></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
and follow it with enough text to ensure that there will be at least one full line below
the matrix.
</p><!--l. 1611--><p class="noindent"><span 
class="cmtt-10">&#x005C;hdotsfor</span><span class="obeylines-h"><span 
class="cmtt-10">{</span></span><span 
class="cmti-10">number</span><span class="obeylines-h"><span 
class="cmtt-10">}</span></span> produces a row of dots in a matrix spanning the given number of
columns: <!--l. 1613--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">        <mrow 
>
<mi 
>W</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A6;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo>  <mfenced separators="" 
open="|"  close="|" ><mtable  
equalrows="false" equalcolumns="false" class="array"><mtr><mtd 
class="array"  columnalign="center"><mfrac><mrow>   <mi 
>&#x03D5;</mi></mrow> 
<mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>&#x03D5;</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>&#x025B;</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac> </mtd> <mtd 
class="array"  columnalign="center">     <mn>0</mn>     </mtd><mtd 
class="array"  columnalign="center"><mo 
class="MathClass-op">&#x2026;</mo></mtd><mtd 
class="array"  columnalign="center">      <mn>0</mn>      </mtd>
</mtr><mtr><mtd 
class="array"  columnalign="center"><mfrac><mrow> <mi 
>&#x03D5;</mi><msub><mrow 
><mi 
>k</mi></mrow><mrow 
><mi 
>n</mi><mn>2</mn></mrow></msub 
></mrow> 
<mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>&#x03D5;</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>&#x025B;</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac> </mtd> <mtd 
class="array"  columnalign="center"><mfrac><mrow>    <mi 
>&#x03D5;</mi></mrow> 
<mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>&#x03D5;</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>&#x025B;</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac> </mtd> <mtd 
class="array"  columnalign="center"><mo 
class="MathClass-op">&#x2026;</mo></mtd><mtd 
class="array"  columnalign="center">      <mn>0</mn>      </mtd>
</mtr><mtr><mtd 
class="array" columnspan="5" columnalign="center"><mrow 
class="multicolumn-center"><mo 
class="MathClass-punc">.</mo> <mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo><mo 
class="MathClass-punc">.</mo></mrow>
</mtd></mtr><mtr><mtd 
class="array"  columnalign="center"><mfrac><mrow> <mi 
>&#x03D5;</mi><msub><mrow 
><mi 
>k</mi></mrow><mrow 
><mi 
>n</mi><mn>1</mn></mrow></msub 
></mrow>
<mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>&#x03D5;</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>&#x025B;</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac></mtd><mtd 
class="array"  columnalign="center"><mfrac><mrow>  <mi 
>&#x03D5;</mi><msub><mrow 
><mi 
>k</mi></mrow><mrow 
><mi 
>n</mi><mn>2</mn></mrow></msub 
></mrow>
<mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>&#x03D5;</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>&#x025B;</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac></mtd><mtd 
class="array"  columnalign="center"><mo 
class="MathClass-op">&#x2026;</mo></mtd><mtd 
class="array"  columnalign="center"><mfrac><mrow>  <mi 
>&#x03D5;</mi><msub><mrow 
><mi 
>k</mi></mrow><mrow 
><mi 
>n</mi><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msub 
></mrow>
<mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>&#x03D5;</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>&#x025B;</mi></mrow><mrow 
><mi 
>n</mi><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac></mtd><mtd 
class="array"  columnalign="center"><mfrac><mrow>     <mi 
>&#x03D5;</mi></mrow>
<mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>&#x03D5;</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>&#x025B;</mi></mrow><mrow 
><mi 
>n</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac></mtd>
</mtr>   <!--*\c@MaxMatrixCols c--></mtable>                                                                                                 </mfenced>
</mrow></math>

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;[W(&#x005C;Phi)=&#x00A0;&#x005C;begin{Vmatrix}
&#x00A0;<br />&#x005C;dfrac&#x005C;varphi{(&#x005C;varphi&#x005F;1,&#x005C;varepsilon&#x005F;1)}&#x0026;0&#x0026;&#x005C;dots&#x0026;0&#x005C;&#x005C;
&#x00A0;<br />&#x005C;dfrac{&#x005C;varphi&#x00A0;k&#x005F;{n2}}{(&#x005C;varphi&#x005F;2,&#x005C;varepsilon&#x005F;1)}&#x0026;
&#x00A0;<br />&#x005C;dfrac&#x005C;varphi{(&#x005C;varphi&#x005F;2,&#x005C;varepsilon&#x005F;2)}&#x0026;&#x005C;dots&#x0026;0&#x005C;&#x005C;
&#x00A0;<br />&#x005C;hdotsfor{5}&#x005C;&#x005C;
&#x00A0;<br />&#x005C;dfrac{&#x005C;varphi&#x00A0;k&#x005F;{n1}}{(&#x005C;varphi&#x005F;n,&#x005C;varepsilon&#x005F;1)}&#x0026;
&#x00A0;<br />&#x005C;dfrac{&#x005C;varphi&#x00A0;k&#x005F;{n2}}{(&#x005C;varphi&#x005F;n,&#x005C;varepsilon&#x005F;2)}&#x0026;&#x005C;dots&#x0026;
&#x00A0;<br />&#x005C;dfrac{&#x005C;varphi&#x00A0;k&#x005F;{n&#x005C;,n-1}}{(&#x005C;varphi&#x005F;n,&#x005C;varepsilon&#x005F;{n-1})}&#x0026;
&#x00A0;<br />&#x005C;dfrac{&#x005C;varphi}{(&#x005C;varphi&#x005F;n,&#x005C;varepsilon&#x005F;n)}
&#x00A0;<br />&#x005C;end{Vmatrix}&#x005C;]
</div>
</td></tr></table>
<!--l. 1635--><p class="indent">   The spacing of the dots can be varied through use of a square-bracket option, for
example, <span class="obeylines-h"><span 
class="cmtt-10">&#x005C;hdotsfor[1.5]{3}</span></span>. The number in square brackets will be used as a
multiplier; the normal value is 1.
</p>
   <h4 class="subsectionHead"><span class="titlemark">9.18</span> <a 
  name="x1-310009.18"></a>The <span 
class="cmtt-10">&#x005C;substack </span>command</h4>
<!--l. 1641--><p class="noindent">The <span 
class="cmtt-10">&#x005C;substack </span>command can be used to produce a multiline subscript or
superscript: for example

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;sum&#x005F;{&#x005C;substack{0&#x005C;le&#x00A0;i&#x005C;le&#x00A0;m&#x005C;&#x005C;&#x00A0;0&#x003C;j&#x003C;n}}&#x00A0;P(i,j)
</div>
</td></tr></table>
<!--l. 1647--><p class="indent">   produces a two-line subscript underneath the sum: </p><table class="equation"><tr><td>
<!--l. 1650--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<msub><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mtable class="subarray-c" rowspacing="0" columnalign="center"><mtr><mtd><mn>0</mn><mo 
class="MathClass-rel">&#x2264;</mo><mi 
>i</mi><mo 
class="MathClass-rel">&#x2264;</mo><mi 
>m</mi>
</mtd></mtr><mtr><mtd><mn>0</mn><mo 
class="MathClass-rel">&#x003C;</mo><mi 
>j</mi><mo 
class="MathClass-rel">&#x003C;</mo><mi 
>n</mi>
     </mtd></mtr>                                                                                                     </mtable></mrow></msub 
><mi 
>P</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>i</mi><mo 
class="MathClass-punc">,</mo><mi 
>j</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mstyle 
    class="label" id="x1-31001r62"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(62)</td></tr></table>
A slightly more generalized form is the <span 
class="cmtt-10">subarray </span>environment which allows you to
specify that each line should be left-aligned instead of centered, as here:
<table class="equation"><tr><td>

<!--l. 1659--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
<msub><mrow 
><mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mtable class="subarray" rowspacing="0" ><mtr><mtd><mn>0</mn><mo 
class="MathClass-rel">&#x2264;</mo><mi 
>i</mi><mo 
class="MathClass-rel">&#x2264;</mo><mi 
>m</mi>
</mtd></mtr><mtr><mtd><mn>0</mn><mo 
class="MathClass-rel">&#x003C;</mo><mi 
>j</mi><mo 
class="MathClass-rel">&#x003C;</mo><mi 
>n</mi>
     </mtd></mtr>                                                                                                     </mtable></mrow></msub 
><mi 
>P</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>i</mi><mo 
class="MathClass-punc">,</mo><mi 
>j</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mstyle 
    class="label" id="x1-31002r63"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(63)</td></tr></table>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;sum&#x005F;{&#x005C;begin{subarray}{l}
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;0&#x005C;le&#x00A0;i&#x005C;le&#x00A0;m&#x005C;&#x005C;&#x00A0;0&#x003C;j&#x003C;n
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x005C;end{subarray}}
&#x00A0;<br />&#x00A0;P(i,j)
</div>
</td></tr></table>
   <h4 class="subsectionHead"><span class="titlemark">9.19</span> <a 
  name="x1-320009.19"></a>Big-g-g delimiters</h4> Here are some big delimiters, first in <span 
class="cmtt-10">&#x005C;normalsize</span>: <!--l. 1670--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">
<mrow 
>
                      <mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mn>0</mn></mrow><mrow 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
>
     </mrow></msubsup 
><msub><mrow 
><mi 
>L</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mi 
>x</mi></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow>
</mrow></math>

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;[&#x005C;biggl(&#x005C;mathbf{E}&#x005F;{y}
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;int&#x005F;0&#x02C6;{t&#x005F;&#x005C;varepsilon}L&#x005F;{x,y&#x02C6;x(s)}&#x005C;varphi(x)&#x005C;,ds
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;biggr)
&#x00A0;<br />&#x005C;]
</div>
</td></tr></table>
<!--l. 1680--><p class="indent">   and now in <span 
class="cmtt-10">&#x005C;Large </span>size: <!--l. 1682--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">         <mrow 
>
                            <mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mn>0</mn></mrow><mrow 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
>
     </mrow></msubsup 
><msub><mrow 
><mi 
>L</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mi 
>x</mi></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mi 
>d</mi><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow>
</mrow></math>

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
{&#x005C;Large
&#x00A0;<br />&#x005C;[&#x005C;biggl(&#x005C;mathbf{E}&#x005F;{y}
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;int&#x005F;0&#x02C6;{t&#x005F;&#x005C;varepsilon}L&#x005F;{x,y&#x02C6;x(s)}&#x005C;varphi(x)&#x005C;,ds
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;biggr)
&#x00A0;<br />&#x005C;]}
</div>
</td></tr></table>

   <h3 class="sectionHead"><span class="titlemark">A</span> <a 
  name="x1-33000A"></a>Examples of multiple-line equation structures</h3>
<!--l. 1724--><p class="noindent"><span 
class="cmbx-12">Note: Starting on this page, vertical rules are added at</span>
<span 
class="cmbx-12">the margins so that the positioning of various display</span>
<span 
class="cmbx-12">elements with respect to the margins can be seen more</span>
<span 
class="cmbx-12">clearly.</span>
</p>
   <h4 class="subsectionHead"><span class="titlemark">A.1</span> <a 
  name="x1-34000A.1"></a>Split</h4> The <span 
class="cmtt-10">split </span>environment is not an independent environment but should
be used inside something else such as <span 
class="cmtt-10">equation </span>or <span 
class="cmtt-10">align</span>.
<!--l. 1731--><p class="noindent">If there is not enough room for it, the equation number for a <span 
class="cmtt-10">split </span>will be shifted to
the previous line, when equation numbers are on the left; the number shifts down to
the next line when numbers are on the right. </p><table class="equation"><tr><td>
                                                                  

<!--l. 1751--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow> <msub><mrow 
><mi 
>f</mi></mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x025B;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>&#x025B;</mi><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow></msub 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
   </mrow><mrow 
><mn>0</mn></mrow><mrow 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
>
     </mrow></msubsup 
><msub><mrow 
><mi 
>L</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>y</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x025B;</mi><mi 
>u</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>u</mi>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>           <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>h</mi><mo 
class="MathClass-op">&#x222B;</mo>
  <msub><mrow 
><mi 
>L</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>z</mi></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>&#x03C1;</mi></mrow><mrow 
><mi 
>x</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>d</mi><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>             <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-bin">+</mo> <mi 
>h</mi><mrow><mo 
class="MathClass-open">[</mo><mrow> <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mn>0</mn></mrow><mrow 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
>
     </mrow></msubsup 
><msub><mrow 
><mi 
>L</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mi 
>x</mi></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>s</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
> <mo 
class="MathClass-op">&#x222B;</mo>
   <msub><mrow 
><mi 
>L</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>z</mi></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>&#x03C1;</mi></mrow><mrow 
><mi 
>x</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>d</mi><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>             <mrow 
class="split-mtd"></mrow>          <mo 
class="MathClass-bin">+</mo> <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mn>0</mn></mrow><mrow 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
>
     </mrow></msubsup 
><msub><mrow 
><mi 
>L</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mi 
>x</mi></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>s</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow></msub 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
   </mrow><mrow 
><mn>0</mn></mrow><mrow 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
>
     </mrow></msubsup 
><msub><mrow 
><mi 
>L</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>y</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x025B;</mi><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">]</mo></mrow>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>             <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>h</mi><msub><mrow 
><mover 
accent="false"><mrow 
><mi 
>L</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>x</mi></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">+</mo> <mi 
>h</mi><msub><mrow 
><mi 
>&#x03B8;</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo>
    </mtd></mtr></mtable>                                                                                     <mstyle 
    class="label" id="x1-34001r64"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(64)</td></tr></table>
Some text after to test the below-display spacing.
                                                                  

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{equation}
&#x00A0;<br />&#x005C;begin{split}
&#x00A0;<br />f&#x005F;{h,&#x005C;varepsilon}(x,y)
&#x00A0;<br />&#x0026;=&#x005C;varepsilon&#x005C;mathbf{E}&#x005F;{x,y}&#x005C;int&#x005F;0&#x02C6;{t&#x005F;&#x005C;varepsilon}
&#x00A0;<br />L&#x005F;{x,y&#x005F;&#x005C;varepsilon(&#x005C;varepsilon&#x00A0;u)}&#x005C;varphi(x)&#x005C;,du&#x005C;&#x005C;
&#x00A0;<br />&#x0026;=&#x00A0;h&#x005C;int&#x00A0;L&#x005F;{x,z}&#x005C;varphi(x)&#x005C;rho&#x005F;x(dz)&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;quad+h&#x005C;biggl[&#x005C;frac{1}{t&#x005F;&#x005C;varepsilon}&#x005C;biggl(&#x005C;mathbf{E}&#x005F;{y}
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;int&#x005F;0&#x02C6;{t&#x005F;&#x005C;varepsilon}L&#x005F;{x,y&#x02C6;x(s)}&#x005C;varphi(x)&#x005C;,ds
&#x00A0;<br />&#x00A0;&#x00A0;-t&#x005F;&#x005C;varepsilon&#x005C;int&#x00A0;L&#x005F;{x,z}&#x005C;varphi(x)&#x005C;rho&#x005F;x(dz)&#x005C;biggr)&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;phantom{{=}+h&#x005C;biggl[}+&#x005C;frac{1}{t&#x005F;&#x005C;varepsilon}
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;biggl(&#x005C;mathbf{E}&#x005F;{y}&#x005C;int&#x005F;0&#x02C6;{t&#x005F;&#x005C;varepsilon}L&#x005F;{x,y&#x02C6;x(s)}
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x005C;varphi(x)&#x005C;,ds&#x00A0;-&#x005C;mathbf{E}&#x005F;{x,y}&#x005C;int&#x005F;0&#x02C6;{t&#x005F;&#x005C;varepsilon}
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;L&#x005F;{x,y&#x005F;&#x005C;varepsilon(&#x005C;varepsilon&#x00A0;s)}
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;&#x005C;varphi(x)&#x005C;,ds&#x005C;biggr)&#x005C;biggr]&#x005C;&#x005C;
&#x00A0;<br />&#x0026;=h&#x005C;wh{L}&#x005F;x&#x005C;varphi(x)+h&#x005C;theta&#x005F;&#x005C;varepsilon(x,y),
&#x00A0;<br />&#x005C;end{split}
&#x00A0;<br />&#x005C;end{equation}
</div>
</td></tr></table>
                                                                  

<!--l. 1776--><p class="indent">   Unnumbered version: </p><table class="equation"><tr><td>
<!--l. 1793--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow> <msub><mrow 
><mi 
>f</mi></mrow><mrow 
><mi 
>h</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x025B;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>&#x025B;</mi><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow></msub 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
   </mrow><mrow 
><mn>0</mn></mrow><mrow 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
>
     </mrow></msubsup 
><msub><mrow 
><mi 
>L</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>y</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x025B;</mi><mi 
>u</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>u</mi>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>           <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>h</mi><mo 
class="MathClass-op">&#x222B;</mo>
  <msub><mrow 
><mi 
>L</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>z</mi></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>&#x03C1;</mi></mrow><mrow 
><mi 
>x</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>d</mi><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>             <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-bin">+</mo> <mi 
>h</mi><mrow><mo 
class="MathClass-open">[</mo><mrow> <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mn>0</mn></mrow><mrow 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
>
     </mrow></msubsup 
><msub><mrow 
><mi 
>L</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mi 
>x</mi></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>s</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
> <mo 
class="MathClass-op">&#x222B;</mo>
   <msub><mrow 
><mi 
>L</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>z</mi></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>&#x03C1;</mi></mrow><mrow 
><mi 
>x</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>d</mi><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>             <mrow 
class="split-mtd"></mrow>          <mo 
class="MathClass-bin">+</mo> <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
></mrow></mfrac><mrow><mo 
class="MathClass-open">(</mo><mrow><msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mn>0</mn></mrow><mrow 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
>
     </mrow></msubsup 
><msub><mrow 
><mi 
>L</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mi 
>x</mi></mrow></msup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>s</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>E</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow></msub 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
   </mrow><mrow 
><mn>0</mn></mrow><mrow 
><msub><mrow 
><mi 
>t</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
>
     </mrow></msubsup 
><msub><mrow 
><mi 
>L</mi></mrow><mrow 
><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>y</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x025B;</mi><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>s</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">]</mo></mrow>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>             <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>h</mi><msub><mrow 
><mover 
accent="false"><mrow 
><mi 
>L</mi></mrow><mo 
class="MathClass-op">&#x0302;</mo></mover></mrow><mrow 
><mi 
>x</mi></mrow></msub 
><mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">+</mo> <mi 
>h</mi><msub><mrow 
><mi 
>&#x03B8;</mi></mrow><mrow 
><mi 
>&#x025B;</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo>
    </mtd></mtr></mtable>                                                                                     
</math>
</td></tr></table>
Some text after to test the below-display spacing.
                                                                  

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{equation*}
&#x00A0;<br />&#x005C;begin{split}
&#x00A0;<br />f&#x005F;{h,&#x005C;varepsilon}(x,y)
&#x00A0;<br />&#x0026;=&#x005C;varepsilon&#x005C;mathbf{E}&#x005F;{x,y}&#x005C;int&#x005F;0&#x02C6;{t&#x005F;&#x005C;varepsilon}
&#x00A0;<br />L&#x005F;{x,y&#x005F;&#x005C;varepsilon(&#x005C;varepsilon&#x00A0;u)}&#x005C;varphi(x)&#x005C;,du&#x005C;&#x005C;
&#x00A0;<br />&#x0026;=&#x00A0;h&#x005C;int&#x00A0;L&#x005F;{x,z}&#x005C;varphi(x)&#x005C;rho&#x005F;x(dz)&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;quad+h&#x005C;biggl[&#x005C;frac{1}{t&#x005F;&#x005C;varepsilon}&#x005C;biggl(&#x005C;mathbf{E}&#x005F;{y}
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;int&#x005F;0&#x02C6;{t&#x005F;&#x005C;varepsilon}L&#x005F;{x,y&#x02C6;x(s)}&#x005C;varphi(x)&#x005C;,ds
&#x00A0;<br />&#x00A0;&#x00A0;-t&#x005F;&#x005C;varepsilon&#x005C;int&#x00A0;L&#x005F;{x,z}&#x005C;varphi(x)&#x005C;rho&#x005F;x(dz)&#x005C;biggr)&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;phantom{{=}+h&#x005C;biggl[}+&#x005C;frac{1}{t&#x005F;&#x005C;varepsilon}
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;biggl(&#x005C;mathbf{E}&#x005F;{y}&#x005C;int&#x005F;0&#x02C6;{t&#x005F;&#x005C;varepsilon}L&#x005F;{x,y&#x02C6;x(s)}
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;&#x00A0;&#x005C;varphi(x)&#x005C;,ds&#x00A0;-&#x005C;mathbf{E}&#x005F;{x,y}&#x005C;int&#x005F;0&#x02C6;{t&#x005F;&#x005C;varepsilon}
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;L&#x005F;{x,y&#x005F;&#x005C;varepsilon(&#x005C;varepsilon&#x00A0;s)}
&#x00A0;<br />&#x00A0;&#x00A0;&#x00A0;&#x005C;varphi(x)&#x005C;,ds&#x005C;biggr)&#x005C;biggr]&#x005C;&#x005C;
&#x00A0;<br />&#x0026;=h&#x005C;wh{L}&#x005F;x&#x005C;varphi(x)+h&#x005C;theta&#x005F;&#x005C;varepsilon(x,y),
&#x00A0;<br />&#x005C;end{split}
&#x00A0;<br />&#x005C;end{equation*}
</div>
</td></tr></table>
                                                                  

<!--l. 1818--><p class="indent">   If the option <span 
class="cmtt-10">centertags </span>is included in the options list of the <span 
class="cmtt-10">amsmath </span>package,
the equation numbers for <span 
class="cmtt-10">split </span>environments will be centered vertically on the
height of the <span 
class="cmtt-10">split</span>: </p><table class="equation"><tr><td>
<!--l. 1834--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
class="equation">     
    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>  <mfenced separators="" 
open="|"  close="|" ><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mfenced><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="|"  close="|" ><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
 </mrow><mrow 
><mn>0</mn></mrow><mrow 
><mi 
>T</mi></mrow></msubsup 
><mi 
>&#x03C8;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mfenced separators="" 
open="{"  close="}" ><mi 
>u</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>a</mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>&#x03B3;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mi 
>a</mi></mrow></msubsup 
>   <mfrac><mrow 
><mi 
>d</mi><mi 
>&#x03B8;</mi></mrow>
<mrow 
><mi 
>k</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03B8;</mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
      </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>&#x03B8;</mi></mrow></msubsup 
><mi 
>c</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BE;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>u</mi></mrow><mrow 
>
<mi 
>t</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BE;</mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>&#x03BE;</mi></mfenced><mi 
>d</mi><mi 
>t</mi></mfenced>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>    <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">&#x2264;</mo> <msub><mrow 
><mi 
>C</mi></mrow><mrow 
><mn>6</mn></mrow></msub 
> <mfenced separators="" 
open="|"  close="|" ><mfenced separators="" 
open="|"  close="|" ><mi 
>f</mi><msub><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>&#x03A9;</mi></mrow></msub 
> <mfenced separators="" 
open="|"  close="|" ><msubsup><mrow 
><mover 
accent="false"><mrow 
><mi 
>S</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover></mrow><mrow 
><mi 
>a</mi><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-bin">&#x2212;</mo></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn><mo 
class="MathClass-punc">,</mo><mn>0</mn></mrow></msubsup 
><msub><mrow 
><mi 
>W</mi></mrow><mrow 
>
<mn>2</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>&#x0393;</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mfenced separators="" 
open="|"  close="|" ><mfenced separators="" 
open="|"  close="|" ><mi 
>u</mi></mfenced><mover><mrow 
> <mo 
class="MathClass-bin">&#x25E6;</mo></mrow><mrow 
><mo 
class="MathClass-rel">&#x2192;</mo></mrow></mover><msubsup><mrow 
><mi 
>W</mi></mrow><mrow 
><mn>2</mn></mrow><mrow 
><mover 
accent="false"><mrow 
><mi 
>A</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover></mrow></msubsup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">;</mo><msub><mrow 
><mi 
>&#x0393;</mi></mrow><mrow 
>
<mi 
>r</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mi 
>T</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced></mfenced><mo 
class="MathClass-punc">.</mo>
    </mtd></mtr></mtable>                                                                                    <mstyle 
    class="label" id="x1-34002r65"  ></mstyle><!--endlabel-->
</math>
</td><td class="eq-no">(65)</td></tr></table>
Some text after to test the below-display spacing.
                                                                  

   Use of <span 
class="cmtt-10">split </span>within <span 
class="cmtt-10">align</span>:
<!--tex4ht:inline--><!--l. 1864--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
display="block"><mtable 
class="align">
        <mtr><mtd 
class="align-odd">    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>  <mfenced separators="" 
open="|"  close="|" ><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
></mfenced><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="|"  close="|" ><msub><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>&#x03A9;</mi></mrow></msub 
><mi 
>g</mi><mi 
>R</mi><mi 
>u</mi><mi 
>d</mi><mi 
>&#x03A9;</mi></mfenced>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>     <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">&#x2264;</mo> <msub><mrow 
><mi 
>C</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
><msup><mrow 
> <mfenced separators="" 
open="["  close="]" ><msub><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>&#x03A9;</mi></mrow></msub 
><msup><mrow 
> <mfenced separators="" 
open="("  close=")" ><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
 </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>x</mi></mrow></msubsup 
><mi 
>g</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BE;</mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>&#x03BE;</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
><mi 
>d</mi><mi 
>&#x03A9;</mi></mfenced></mrow><mrow 
><mn>1</mn><mo 
class="MathClass-bin">/</mo><mn>2</mn></mrow></msup 
>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>      <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-bin">&#x00D7;</mo><msup><mrow 
><mfenced separators="" 
open="["  close="]" ><msub><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>&#x03A9;</mi></mrow></msub 
> <mfenced separators="" 
open="{"  close="}" ><msubsup><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>x</mi></mrow><mrow 
><mn>2</mn></mrow></msubsup 
> <mo 
class="MathClass-bin">+</mo> <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><mi 
>k</mi></mrow></mfrac><msup><mrow 
> <mfenced separators="" 
open="("  close=")" ><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
 </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>x</mi></mrow></msubsup 
><mi 
>c</mi><msub><mrow 
><mi 
>u</mi></mrow><mrow 
>
<mi 
>t</mi></mrow></msub 
> <mi 
>d</mi><mi 
>&#x03BE;</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
></mfenced> <mi 
>c</mi><mi 
>&#x03A9;</mi></mfenced></mrow><mrow 
><mn>1</mn><mo 
class="MathClass-bin">/</mo><mn>2</mn></mrow></msup 
>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>      <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">&#x2264;</mo> <msub><mrow 
><mi 
>C</mi></mrow><mrow 
><mn>4</mn></mrow></msub 
> <mfenced separators="" 
open="|"  close="|" ><mfenced separators="" 
open="|"  close="|" ><mi 
>f</mi> <mfenced separators="" 
open="|"  close="|" ><msubsup><mrow 
><mover 
accent="false"><mrow 
><mi 
>S</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover></mrow><mrow 
><mi 
>a</mi><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-bin">&#x2212;</mo></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn><mo 
class="MathClass-punc">,</mo><mn>0</mn></mrow></msubsup 
><msub><mrow 
><mi 
>W</mi></mrow><mrow 
>
<mn>2</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>&#x0393;</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mfenced separators="" 
open="|"  close="|" ><mfenced separators="" 
open="|"  close="|" ><mi 
>u</mi></mfenced><mover><mrow 
> <mo 
class="MathClass-bin">&#x25E6;</mo></mrow><mrow 
><mo 
class="MathClass-rel">&#x2192;</mo></mrow></mover><msubsup><mrow 
><mi 
>W</mi></mrow><mrow 
><mn>2</mn></mrow><mrow 
><mover 
accent="false"><mrow 
><mi 
>A</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover></mrow></msubsup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">;</mo><msub><mrow 
><mi 
>&#x0393;</mi></mrow><mrow 
>
<mi 
>r</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mi 
>T</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced></mfenced><mo 
class="MathClass-punc">.</mo>
    </mtd></mtr></mtable>                                                                          </mtd>        <mtd 
class="align-even"></mtd>        <mtd 
class="align-label">      <mstyle 
    class="label" id="x1-34003r66"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>6</mn><mn>6</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
        </mtd></mtr><mtr><mtd 
class="align-odd">    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>  <mfenced separators="" 
open="|"  close="|" ><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mfenced><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="|"  close="|" ><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
 </mrow><mrow 
><mn>0</mn></mrow><mrow 
><mi 
>T</mi></mrow></msubsup 
><mi 
>&#x03C8;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mfenced separators="" 
open="{"  close="}" ><mi 
>u</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>a</mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>&#x03B3;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mi 
>a</mi></mrow></msubsup 
>   <mfrac><mrow 
><mi 
>d</mi><mi 
>&#x03B8;</mi></mrow>
<mrow 
><mi 
>k</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03B8;</mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
      </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>&#x03B8;</mi></mrow></msubsup 
><mi 
>c</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BE;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>u</mi></mrow><mrow 
>
<mi 
>t</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BE;</mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>&#x03BE;</mi></mfenced><mi 
>d</mi><mi 
>t</mi></mfenced>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>    <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">&#x2264;</mo> <msub><mrow 
><mi 
>C</mi></mrow><mrow 
><mn>6</mn></mrow></msub 
> <mfenced separators="" 
open="|"  close="|" ><mfenced separators="" 
open="|"  close="|" ><mi 
>f</mi><msub><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>&#x03A9;</mi></mrow></msub 
> <mfenced separators="" 
open="|"  close="|" ><msubsup><mrow 
><mover 
accent="false"><mrow 
><mi 
>S</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover></mrow><mrow 
><mi 
>a</mi><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-bin">&#x2212;</mo></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn><mo 
class="MathClass-punc">,</mo><mn>0</mn></mrow></msubsup 
><msub><mrow 
><mi 
>W</mi></mrow><mrow 
>
<mn>2</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>&#x0393;</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mfenced separators="" 
open="|"  close="|" ><mfenced separators="" 
open="|"  close="|" ><mi 
>u</mi></mfenced><mover><mrow 
> <mo 
class="MathClass-bin">&#x25E6;</mo></mrow><mrow 
><mo 
class="MathClass-rel">&#x2192;</mo></mrow></mover><msubsup><mrow 
><mi 
>W</mi></mrow><mrow 
><mn>2</mn></mrow><mrow 
><mover 
accent="false"><mrow 
><mi 
>A</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover></mrow></msubsup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">;</mo><msub><mrow 
><mi 
>&#x0393;</mi></mrow><mrow 
>
<mi 
>r</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mi 
>T</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced></mfenced><mo 
class="MathClass-punc">.</mo>
    </mtd></mtr></mtable>                                                                          </mtd>        <mtd 
class="align-even"></mtd>        <mtd 
class="align-label">      <mstyle 
    class="label" id="x1-34004r67"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>6</mn><mn>7</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
   </mtd></mtr></mtable></math>
Some text after to test the below-display spacing.
                                                                  

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{align}
&#x00A0;<br />&#x005C;begin{split}&#x005C;abs{I&#x005F;1}
&#x00A0;<br />&#x00A0;&#x00A0;&#x0026;=&#x005C;left&#x005C;lvert&#x00A0;&#x005C;int&#x005F;&#x005C;Omega&#x00A0;gRu&#x005C;,d&#x005C;Omega&#x005C;right&#x005C;rvert&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;le&#x00A0;C&#x005F;3&#x005C;left[&#x005C;int&#x005F;&#x005C;Omega&#x005C;left(&#x005C;int&#x005F;{a}&#x02C6;x
&#x00A0;<br />&#x00A0;&#x00A0;g(&#x005C;xi,t)&#x005C;,d&#x005C;xi&#x005C;right)&#x02C6;2d&#x005C;Omega&#x005C;right]&#x02C6;{1/2}&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;quad&#x005C;times&#x00A0;&#x005C;left[&#x005C;int&#x005F;&#x005C;Omega&#x005C;left&#x005C;{u&#x02C6;2&#x005F;x+&#x005C;frac{1}{k}
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;left(&#x005C;int&#x005F;{a}&#x02C6;x&#x00A0;cu&#x005F;t&#x005C;,d&#x005C;xi&#x005C;right)&#x02C6;2&#x005C;right&#x005C;}
&#x00A0;<br />&#x00A0;&#x00A0;c&#x005C;Omega&#x005C;right]&#x02C6;{1/2}&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;le&#x00A0;C&#x005F;4&#x005C;left&#x005C;lvert&#x00A0;&#x005C;left&#x005C;lvert&#x00A0;f&#x005C;left&#x005C;lvert&#x00A0;&#x005C;wt{S}&#x02C6;{-1,0}&#x005F;{a,-}
&#x00A0;<br />&#x00A0;&#x00A0;W&#x005F;2(&#x005C;Omega,&#x005C;Gamma&#x005F;l)&#x005C;right&#x005C;rvert&#x005C;right&#x005C;rvert
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;left&#x005C;lvert&#x00A0;&#x005C;abs{u}&#x005C;overset{&#x005C;circ}&#x005C;to&#x00A0;W&#x005F;2&#x02C6;{&#x005C;wt{A}}
&#x00A0;<br />&#x00A0;&#x00A0;(&#x005C;Omega;&#x005C;Gamma&#x005F;r,T)&#x005C;right&#x005C;rvert&#x005C;right&#x005C;rvert.
&#x00A0;<br />&#x005C;end{split}&#x005C;label{eq:A}&#x005C;&#x005C;
&#x00A0;<br />&#x005C;begin{split}&#x005C;abs{I&#x005F;2}&#x0026;=&#x005C;left&#x005C;lvert&#x00A0;&#x005C;int&#x005F;{0}&#x02C6;T&#x00A0;&#x005C;psi(t)&#x005C;left&#x005C;{u(a,t)
&#x00A0;<br />&#x00A0;&#x00A0;-&#x005C;int&#x005F;{&#x005C;gamma(t)}&#x02C6;a&#x005C;frac{d&#x005C;theta}{k(&#x005C;theta,t)}
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;int&#x005F;{a}&#x02C6;&#x005C;theta&#x00A0;c(&#x005C;xi)u&#x005F;t(&#x005C;xi,t)&#x005C;,d&#x005C;xi&#x005C;right&#x005C;}dt&#x005C;right&#x005C;rvert&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;le&#x00A0;C&#x005F;6&#x005C;left&#x005C;lvert&#x00A0;&#x005C;left&#x005C;lvert&#x00A0;f&#x005C;int&#x005F;&#x005C;Omega
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;left&#x005C;lvert&#x00A0;&#x005C;wt{S}&#x02C6;{-1,0}&#x005F;{a,-}
&#x00A0;<br />&#x00A0;&#x00A0;W&#x005F;2(&#x005C;Omega,&#x005C;Gamma&#x005F;l)&#x005C;right&#x005C;rvert&#x005C;right&#x005C;rvert
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;left&#x005C;lvert&#x00A0;&#x005C;abs{u}&#x005C;overset{&#x005C;circ}&#x005C;to&#x00A0;W&#x005F;2&#x02C6;{&#x005C;wt{A}}
&#x00A0;<br />&#x00A0;&#x00A0;(&#x005C;Omega;&#x005C;Gamma&#x005F;r,T)&#x005C;right&#x005C;rvert&#x005C;right&#x005C;rvert.
&#x00A0;<br />&#x005C;end{split}
&#x00A0;<br />&#x005C;end{align}
</div>
</td></tr></table>
                                                                  

<!--l. 1896--><p class="indent">   Unnumbered <span 
class="cmtt-10">align</span>, with a number on the second <span 
class="cmtt-10">split</span>:
<!--tex4ht:inline--></p><!--l. 1918--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
display="block"><mtable 
class="align-star">
       <mtr><mtd 
class="align-odd">    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>  <mfenced separators="" 
open="|"  close="|" ><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mn>1</mn></mrow></msub 
></mfenced><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="|"  close="|" ><msub><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>&#x03A9;</mi></mrow></msub 
><mi 
>g</mi><mi 
>R</mi><mi 
>u</mi><mi 
>d</mi><mi 
>&#x03A9;</mi></mfenced>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>     <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">&#x2264;</mo> <msub><mrow 
><mi 
>C</mi></mrow><mrow 
><mn>3</mn></mrow></msub 
><msup><mrow 
> <mfenced separators="" 
open="["  close="]" ><msub><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>&#x03A9;</mi></mrow></msub 
><msup><mrow 
> <mfenced separators="" 
open="("  close=")" ><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
 </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>x</mi></mrow></msubsup 
><mi 
>g</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BE;</mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>&#x03BE;</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
><mi 
>d</mi><mi 
>&#x03A9;</mi></mfenced></mrow><mrow 
><mn>1</mn><mo 
class="MathClass-bin">/</mo><mn>2</mn></mrow></msup 
>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>      <mrow 
class="split-mtd"></mrow>    <mo 
class="MathClass-bin">&#x00D7;</mo><msup><mrow 
><mfenced separators="" 
open="["  close="]" ><msub><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>&#x03A9;</mi></mrow></msub 
> <mfenced separators="" 
open="{"  close="}" ><msubsup><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>x</mi></mrow><mrow 
><mn>2</mn></mrow></msubsup 
> <mo 
class="MathClass-bin">+</mo> <mfrac><mrow 
><mn>1</mn></mrow> 
<mrow 
><mi 
>k</mi></mrow></mfrac><msup><mrow 
> <mfenced separators="" 
open="("  close=")" ><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
 </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>x</mi></mrow></msubsup 
><mi 
>c</mi><msub><mrow 
><mi 
>u</mi></mrow><mrow 
>
<mi 
>t</mi></mrow></msub 
> <mi 
>d</mi><mi 
>&#x03BE;</mi></mfenced></mrow><mrow 
><mn>2</mn></mrow></msup 
></mfenced> <mi 
>c</mi><mi 
>&#x03A9;</mi></mfenced></mrow><mrow 
><mn>1</mn><mo 
class="MathClass-bin">/</mo><mn>2</mn></mrow></msup 
>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>      <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">&#x2264;</mo> <msub><mrow 
><mi 
>C</mi></mrow><mrow 
><mn>4</mn></mrow></msub 
> <mfenced separators="" 
open="|"  close="|" ><mfenced separators="" 
open="|"  close="|" ><mi 
>f</mi> <mfenced separators="" 
open="|"  close="|" ><msubsup><mrow 
><mover 
accent="false"><mrow 
><mi 
>S</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover></mrow><mrow 
><mi 
>a</mi><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-bin">&#x2212;</mo></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn><mo 
class="MathClass-punc">,</mo><mn>0</mn></mrow></msubsup 
><msub><mrow 
><mi 
>W</mi></mrow><mrow 
>
<mn>2</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>&#x0393;</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mfenced separators="" 
open="|"  close="|" ><mfenced separators="" 
open="|"  close="|" ><mi 
>u</mi></mfenced><mover><mrow 
> <mo 
class="MathClass-bin">&#x25E6;</mo></mrow><mrow 
><mo 
class="MathClass-rel">&#x2192;</mo></mrow></mover><msubsup><mrow 
><mi 
>W</mi></mrow><mrow 
><mn>2</mn></mrow><mrow 
><mover 
accent="false"><mrow 
><mi 
>A</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover></mrow></msubsup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">;</mo><msub><mrow 
><mi 
>&#x0393;</mi></mrow><mrow 
>
<mi 
>r</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mi 
>T</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced></mfenced><mo 
class="MathClass-punc">.</mo>
    </mtd></mtr></mtable>                                                                          </mtd>
       </mtr><mtr><mtd 
class="align-odd">    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>  <mfenced separators="" 
open="|"  close="|" ><msub><mrow 
><mi 
>I</mi></mrow><mrow 
><mn>2</mn></mrow></msub 
></mfenced><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="|"  close="|" ><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
 </mrow><mrow 
><mn>0</mn></mrow><mrow 
><mi 
>T</mi></mrow></msubsup 
><mi 
>&#x03C8;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mfenced separators="" 
open="{"  close="}" ><mi 
>u</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>a</mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>&#x03B3;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mi 
>a</mi></mrow></msubsup 
>   <mfrac><mrow 
><mi 
>d</mi><mi 
>&#x03B8;</mi></mrow>
<mrow 
><mi 
>k</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03B8;</mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></mfrac><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
      </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>&#x03B8;</mi></mrow></msubsup 
><mi 
>c</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BE;</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msub><mrow 
><mi 
>u</mi></mrow><mrow 
>
<mi 
>t</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03BE;</mi><mo 
class="MathClass-punc">,</mo><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>&#x03BE;</mi></mfenced><mi 
>d</mi><mi 
>t</mi></mfenced>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>    <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">&#x2264;</mo> <msub><mrow 
><mi 
>C</mi></mrow><mrow 
><mn>6</mn></mrow></msub 
> <mfenced separators="" 
open="|"  close="|" ><mfenced separators="" 
open="|"  close="|" ><mi 
>f</mi><msub><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>&#x03A9;</mi></mrow></msub 
> <mfenced separators="" 
open="|"  close="|" ><msubsup><mrow 
><mover 
accent="false"><mrow 
><mi 
>S</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover></mrow><mrow 
><mi 
>a</mi><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-bin">&#x2212;</mo></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn><mo 
class="MathClass-punc">,</mo><mn>0</mn></mrow></msubsup 
><msub><mrow 
><mi 
>W</mi></mrow><mrow 
>
<mn>2</mn></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">,</mo><msub><mrow 
><mi 
>&#x0393;</mi></mrow><mrow 
><mi 
>l</mi></mrow></msub 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mfenced separators="" 
open="|"  close="|" ><mfenced separators="" 
open="|"  close="|" ><mi 
>u</mi></mfenced><mover><mrow 
> <mo 
class="MathClass-bin">&#x25E6;</mo></mrow><mrow 
><mo 
class="MathClass-rel">&#x2192;</mo></mrow></mover><msubsup><mrow 
><mi 
>W</mi></mrow><mrow 
><mn>2</mn></mrow><mrow 
><mover 
accent="false"><mrow 
><mi 
>A</mi></mrow><mo 
class="MathClass-op">&#x02DC;</mo></mover></mrow></msubsup 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>&#x03A9;</mi><mo 
class="MathClass-punc">;</mo><msub><mrow 
><mi 
>&#x0393;</mi></mrow><mrow 
>
<mi 
>r</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mi 
>T</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mfenced></mfenced><mo 
class="MathClass-punc">.</mo>
    </mtd></mtr></mtable>                                                                          </mtd>       <mtd 
class="align-even"></mtd>       <mtd 
class="align-label">     <mrow><mo 
class="MathClass-open">(</mo><mrow><mn>6</mn><msup><mrow 
><mn>7</mn></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow><mstyle 
    class="label" id="x1-34006r67"  ></mstyle><!--endlabel-->
   </mtd></mtr></mtable></math>
Some text after to test the below-display spacing.
                                                                  

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{align*}
&#x00A0;<br />&#x005C;begin{split}&#x005C;abs{I&#x005F;1}&#x0026;=&#x005C;left&#x005C;lvert&#x00A0;&#x005C;int&#x005F;&#x005C;Omega&#x00A0;gRu&#x005C;,d&#x005C;Omega&#x005C;right&#x005C;rvert&#x005C;&#x005C;
&#x00A0;<br />&#x00A0;&#x00A0;&#x0026;&#x005C;le&#x00A0;C&#x005F;3&#x005C;left[&#x005C;int&#x005F;&#x005C;Omega&#x005C;left(&#x005C;int&#x005F;{a}&#x02C6;x
&#x00A0;<br />&#x00A0;&#x00A0;g(&#x005C;xi,t)&#x005C;,d&#x005C;xi&#x005C;right)&#x02C6;2d&#x005C;Omega&#x005C;right]&#x02C6;{1/2}&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;phantom{=}&#x005C;times&#x00A0;&#x005C;left[&#x005C;int&#x005F;&#x005C;Omega&#x005C;left&#x005C;{u&#x02C6;2&#x005F;x+&#x005C;frac{1}{k}
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;left(&#x005C;int&#x005F;{a}&#x02C6;x&#x00A0;cu&#x005F;t&#x005C;,d&#x005C;xi&#x005C;right)&#x02C6;2&#x005C;right&#x005C;}
&#x00A0;<br />&#x00A0;&#x00A0;c&#x005C;Omega&#x005C;right]&#x02C6;{1/2}&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;le&#x00A0;C&#x005F;4&#x005C;left&#x005C;lvert&#x00A0;&#x005C;left&#x005C;lvert&#x00A0;f&#x005C;left&#x005C;lvert&#x00A0;&#x005C;wt{S}&#x02C6;{-1,0}&#x005F;{a,-}
&#x00A0;<br />&#x00A0;&#x00A0;W&#x005F;2(&#x005C;Omega,&#x005C;Gamma&#x005F;l)&#x005C;right&#x005C;rvert&#x005C;right&#x005C;rvert
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;left&#x005C;lvert&#x00A0;&#x005C;abs{u}&#x005C;overset{&#x005C;circ}&#x005C;to&#x00A0;W&#x005F;2&#x02C6;{&#x005C;wt{A}}
&#x00A0;<br />&#x00A0;&#x00A0;(&#x005C;Omega;&#x005C;Gamma&#x005F;r,T)&#x005C;right&#x005C;rvert&#x005C;right&#x005C;rvert.
&#x00A0;<br />&#x005C;end{split}&#x005C;&#x005C;
&#x00A0;<br />&#x005C;begin{split}&#x005C;abs{I&#x005F;2}&#x0026;=&#x005C;left&#x005C;lvert&#x00A0;&#x005C;int&#x005F;{0}&#x02C6;T&#x00A0;&#x005C;psi(t)&#x005C;left&#x005C;{u(a,t)
&#x00A0;<br />&#x00A0;&#x00A0;-&#x005C;int&#x005F;{&#x005C;gamma(t)}&#x02C6;a&#x005C;frac{d&#x005C;theta}{k(&#x005C;theta,t)}
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;int&#x005F;{a}&#x02C6;&#x005C;theta&#x00A0;c(&#x005C;xi)u&#x005F;t(&#x005C;xi,t)&#x005C;,d&#x005C;xi&#x005C;right&#x005C;}dt&#x005C;right&#x005C;rvert&#x005C;&#x005C;
&#x00A0;<br />&#x0026;&#x005C;le&#x00A0;C&#x005F;6&#x005C;left&#x005C;lvert&#x00A0;&#x005C;left&#x005C;lvert&#x00A0;f&#x005C;int&#x005F;&#x005C;Omega
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;left&#x005C;lvert&#x00A0;&#x005C;wt{S}&#x02C6;{-1,0}&#x005F;{a,-}
&#x00A0;<br />&#x00A0;&#x00A0;W&#x005F;2(&#x005C;Omega,&#x005C;Gamma&#x005F;l)&#x005C;right&#x005C;rvert&#x005C;right&#x005C;rvert
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;left&#x005C;lvert&#x00A0;&#x005C;abs{u}&#x005C;overset{&#x005C;circ}&#x005C;to&#x00A0;W&#x005F;2&#x02C6;{&#x005C;wt{A}}
&#x00A0;<br />&#x00A0;&#x00A0;(&#x005C;Omega;&#x005C;Gamma&#x005F;r,T)&#x005C;right&#x005C;rvert&#x005C;right&#x005C;rvert.
&#x00A0;<br />&#x005C;end{split}&#x005C;tag{&#x005C;theequation$&#x2019;$}
&#x00A0;<br />&#x005C;end{align*}
</div>
</td></tr></table>
                                                                  

   <h4 class="subsectionHead"><span class="titlemark">A.2</span> <a 
  name="x1-35000A.2"></a>Multline</h4> Numbered version:
<!--l. 1950--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     
<mtable 
class="multline">
<mtr><mtd 
class="multline"></mtd><mtd><mstyle 
    class="label" id="x1-35001r68"  ></mstyle><!--endlabel--><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>b</mi></mrow></msubsup 
><mrow><mo 
class="MathClass-open">{</mo><mrow><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>b</mi></mrow></msubsup 
><mrow><mo 
class="MathClass-open">[</mo><mrow><mi 
>f</mi><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
><mi 
>g</mi><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
> <mo 
class="MathClass-bin">+</mo> <mi 
>f</mi><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
><mi 
>g</mi><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
></mrow><mo 
class="MathClass-close">]</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>g</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>g</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>x</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mi 
>d</mi><mi 
>y</mi>
</mtd></mtr><mtr><mtd 
class="multline"></mtd><mtd> <mo 
class="MathClass-rel">=</mo><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>b</mi></mrow></msubsup 
><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>g</mi><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>b</mi></mrow></msubsup 
><msup><mrow 
><mi 
>f</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
> <mo 
class="MathClass-bin">+</mo> <mi 
>f</mi><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>b</mi></mrow></msubsup 
><msup><mrow 
><mi 
>g</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>g</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>b</mi></mrow></msubsup 
><mi 
>f</mi><mi 
>g</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mi 
>d</mi><mi 
>y</mi></mtd><mtd><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>6</mn><mn>8</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>                </mtd></mtr></mtable>
</math>
<!--l. 1955--><p class="nopar">
To test the use of <span class="obeylines-h"><span 
class="cmtt-10">&#x005C;label</span></span> and <span class="obeylines-h"><span 
class="cmtt-10">&#x005C;ref</span></span>, we refer to the number of this equation here:
(<a 
href="#x1-35001r68">68<!--tex4ht:ref: eq:E--></a>).
                                                                  

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{multline}&#x005C;label{eq:E}
&#x00A0;<br />&#x005C;int&#x005F;a&#x02C6;b&#x005C;biggl&#x005C;{&#x005C;int&#x005F;a&#x02C6;b[f(x)&#x02C6;2g(y)&#x02C6;2+f(y)&#x02C6;2g(x)&#x02C6;2]
&#x00A0;<br />&#x00A0;-2f(x)g(x)f(y)g(y)&#x005C;,dx&#x005C;biggr&#x005C;}&#x005C;,dy&#x00A0;&#x005C;&#x005C;
&#x00A0;<br />&#x00A0;=&#x005C;int&#x005F;a&#x02C6;b&#x005C;biggl&#x005C;{g(y)&#x02C6;2&#x005C;int&#x005F;a&#x02C6;bf&#x02C6;2+f(y)&#x02C6;2
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;int&#x005F;a&#x02C6;b&#x00A0;g&#x02C6;2-2f(y)g(y)&#x005C;int&#x005F;a&#x02C6;b&#x00A0;fg&#x005C;biggr&#x005C;}&#x005C;,dy
&#x00A0;<br />&#x005C;end{multline}
</div>
</td></tr></table>
<!--l. 1970--><p class="indent">   Unnumbered version:
</p><!--l. 1971--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     
<mtable 
class="multline-star">
<mtr><mtd 
class="multline-star"><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>b</mi></mrow></msubsup 
><mrow><mo 
class="MathClass-open">{</mo><mrow><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>b</mi></mrow></msubsup 
><mrow><mo 
class="MathClass-open">[</mo><mrow><mi 
>f</mi><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
><mi 
>g</mi><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
> <mo 
class="MathClass-bin">+</mo> <mi 
>f</mi><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
><mi 
>g</mi><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
></mrow><mo 
class="MathClass-close">]</mo></mrow> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>g</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>g</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>d</mi><mi 
>x</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mi 
>d</mi><mi 
>y</mi>
</mtd></mtr><mtr><mtd 
class="multline-star"> <mo 
class="MathClass-rel">=</mo><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>b</mi></mrow></msubsup 
><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>g</mi><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>b</mi></mrow></msubsup 
><msup><mrow 
><mi 
>f</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
> <mo 
class="MathClass-bin">+</mo> <mi 
>f</mi><msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
><msubsup><mrow 
> <mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>b</mi></mrow></msubsup 
><msup><mrow 
><mi 
>g</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
> <mo 
class="MathClass-bin">&#x2212;</mo> <mn>2</mn><mi 
>f</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mi 
>g</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><msubsup><mrow 
><mo 
class="MathClass-op">&#x222B;</mo>
  </mrow><mrow 
><mi 
>a</mi></mrow><mrow 
><mi 
>b</mi></mrow></msubsup 
><mi 
>f</mi><mi 
>g</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mi 
>d</mi><mi 
>y</mi>                  </mtd></mtr></mtable>
</math>
<!--l. 1976--><p class="nopar">
Some text after to test the below-display spacing.
                                                                  

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{multline*}
&#x00A0;<br />&#x005C;int&#x005F;a&#x02C6;b&#x005C;biggl&#x005C;{&#x005C;int&#x005F;a&#x02C6;b[f(x)&#x02C6;2g(y)&#x02C6;2+f(y)&#x02C6;2g(x)&#x02C6;2]
&#x00A0;<br />&#x00A0;-2f(x)g(x)f(y)g(y)&#x005C;,dx&#x005C;biggr&#x005C;}&#x005C;,dy&#x00A0;&#x005C;&#x005C;
&#x00A0;<br />&#x00A0;=&#x005C;int&#x005F;a&#x02C6;b&#x005C;biggl&#x005C;{g(y)&#x02C6;2&#x005C;int&#x005F;a&#x02C6;bf&#x02C6;2+f(y)&#x02C6;2
&#x00A0;<br />&#x00A0;&#x00A0;&#x005C;int&#x005F;a&#x02C6;b&#x00A0;g&#x02C6;2-2f(y)g(y)&#x005C;int&#x005F;a&#x02C6;b&#x00A0;fg&#x005C;biggr&#x005C;}&#x005C;,dy
&#x00A0;<br />&#x005C;end{multline*}
</div>
</td></tr></table>
                                                                  

   <h4 class="subsectionHead"><span class="titlemark">A.3</span> <a 
  name="x1-36000A.3"></a>Gather</h4> Numbered version with <span class="obeylines-h"><span 
class="cmtt-10">&#x005C;notag</span></span> on the second line:
<!--l. 2042--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     
<mtable 
class="gather">
<mtr> 
<mtd><mi 
>D</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>a</mi><mo 
class="MathClass-punc">,</mo><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2261;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>z</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>C</mi><mo 
class="MathClass-punc">:</mo>  <mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>a</mi></mfenced> <mo 
class="MathClass-rel">&#x003C;</mo> <mi 
>r</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mo 
class="MathClass-punc">,</mo></mtd>                        
<mtd><mstyle 
    class="label" id="x1-36001r69"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>6</mn><mn>9</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd>
</mtr><mtr> 
<mtd><mo 
class="MathClass-op">seg</mo><!--nolimits--><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>a</mi><mo 
class="MathClass-punc">,</mo><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2261;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>z</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>C</mi><mo 
class="MathClass-punc">:</mo> <mi 
>&#x2111;</mi><mi 
>z</mi> <mo 
class="MathClass-rel">=</mo> <mi 
>&#x2111;</mi><mi 
>a</mi><mo 
class="MathClass-punc">,</mo><mspace class="nbsp" /> <mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>a</mi></mfenced> <mo 
class="MathClass-rel">&#x003C;</mo> <mi 
>r</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mo 
class="MathClass-punc">,</mo></mtd>            
<mtd></mtd>
</mtr><mtr> 
<mtd><mi 
>c</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>e</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03B8;</mi><mo 
class="MathClass-punc">,</mo><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2261;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>C</mi><mo 
class="MathClass-punc">:</mo>  <mfenced separators="" 
open="|"  close="|" ><mi 
>x</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>e</mi></mfenced> <mo 
class="MathClass-rel">&#x003C;</mo> <mi 
>y</mi><mo 
>tan</mo><!--nolimits--><mi 
>&#x03B8;</mi><mo 
class="MathClass-punc">,</mo><mspace class="nbsp" /><mn>0</mn> <mo 
class="MathClass-rel">&#x003C;</mo> <mi 
>y</mi> <mo 
class="MathClass-rel">&#x003C;</mo> <mi 
>r</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mo 
class="MathClass-punc">,</mo></mtd> 
<mtd><mstyle 
    class="label" id="x1-36002r70"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>7</mn><mn>0</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd>
</mtr><mtr> 
<mtd><mi 
>C</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>E</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03B8;</mi><mo 
class="MathClass-punc">,</mo><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2261;</mo><msub><mrow 
><mo 
class="MathClass-op">&#x22C3;</mo>
  </mrow><mrow 
><mi 
>e</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>E</mi></mrow></msub 
><mi 
>c</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>e</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03B8;</mi><mo 
class="MathClass-punc">,</mo><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">.</mo></mtd>                            
<mtd><mstyle 
    class="label" id="x1-36003r71"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>7</mn><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd>                              </mtr></mtable>
</math>
<!--l. 2049--><p class="nopar">
                                                                  

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{gather}
&#x00A0;<br />D(a,r)&#x005C;equiv&#x005C;{z&#x005C;in&#x005C;mathbf{C}&#x005C;colon&#x00A0;&#x005C;abs{z-a}&#x003C;r&#x005C;},&#x005C;&#x005C;
&#x00A0;<br />&#x005C;seg(a,r)&#x005C;equiv&#x005C;{z&#x005C;in&#x005C;mathbf{C}&#x005C;colon
&#x00A0;<br />&#x005C;Im&#x00A0;z=&#x00A0;&#x005C;Im&#x00A0;a,&#x005C;&#x00A0;&#x005C;abs{z-a}&#x003C;r&#x005C;},&#x005C;notag&#x005C;&#x005C;
&#x00A0;<br />c(e,&#x005C;theta,r)&#x005C;equiv&#x005C;{(x,y)&#x005C;in&#x005C;mathbf{C}
&#x00A0;<br />&#x005C;colon&#x00A0;&#x005C;abs{x-e}&#x003C;y&#x005C;tan&#x005C;theta,&#x005C;&#x00A0;0&#x003C;y&#x003C;r&#x005C;},&#x005C;&#x005C;
&#x00A0;<br />C(E,&#x005C;theta,r)&#x005C;equiv&#x005C;bigcup&#x005F;{e&#x005C;in&#x00A0;E}c(e,&#x005C;theta,r).
&#x00A0;<br />&#x005C;end{gather}
</div>
</td></tr></table>
<!--l. 2062--><p class="indent">   Unnumbered version.
</p><!--l. 2063--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     
<mtable 
class="gather-star">
<mtr> 
<mtd><mi 
>D</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>a</mi><mo 
class="MathClass-punc">,</mo><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2261;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>z</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>C</mi><mo 
class="MathClass-punc">:</mo>  <mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>a</mi></mfenced> <mo 
class="MathClass-rel">&#x003C;</mo> <mi 
>r</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mo 
class="MathClass-punc">,</mo></mtd>                        
<mtd></mtd>
</mtr><mtr> 
<mtd><mo 
class="MathClass-op">seg</mo><!--nolimits--><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>a</mi><mo 
class="MathClass-punc">,</mo><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2261;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mi 
>z</mi> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>C</mi><mo 
class="MathClass-punc">:</mo> <mi 
>&#x2111;</mi><mi 
>z</mi> <mo 
class="MathClass-rel">=</mo> <mi 
>&#x2111;</mi><mi 
>a</mi><mo 
class="MathClass-punc">,</mo><mspace class="nbsp" /> <mfenced separators="" 
open="|"  close="|" ><mi 
>z</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>a</mi></mfenced> <mo 
class="MathClass-rel">&#x003C;</mo> <mi 
>r</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mo 
class="MathClass-punc">,</mo></mtd>            
<mtd></mtd>
</mtr><mtr> 
<mtd><mi 
>c</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>e</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03B8;</mi><mo 
class="MathClass-punc">,</mo><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2261;</mo><mrow><mo 
class="MathClass-open">{</mo><mrow><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2208;</mo> <mi 
>C</mi><mo 
class="MathClass-punc">:</mo>  <mfenced separators="" 
open="|"  close="|" ><mi 
>x</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>e</mi></mfenced> <mo 
class="MathClass-rel">&#x003C;</mo> <mi 
>y</mi><mo 
>tan</mo><!--nolimits--><mi 
>&#x03B8;</mi><mo 
class="MathClass-punc">,</mo><mspace class="nbsp" /><mn>0</mn> <mo 
class="MathClass-rel">&#x003C;</mo> <mi 
>y</mi> <mo 
class="MathClass-rel">&#x003C;</mo> <mi 
>r</mi></mrow><mo 
class="MathClass-close">}</mo></mrow><mo 
class="MathClass-punc">,</mo></mtd> 
<mtd></mtd>
</mtr><mtr> 
<mtd><mi 
>C</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>E</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03B8;</mi><mo 
class="MathClass-punc">,</mo><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">&#x2261;</mo><msub><mrow 
><mo 
class="MathClass-op">&#x22C3;</mo>
  </mrow><mrow 
><mi 
>e</mi><mo 
class="MathClass-rel">&#x2208;</mo><mi 
>E</mi></mrow></msub 
><mi 
>c</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>e</mi><mo 
class="MathClass-punc">,</mo><mi 
>&#x03B8;</mi><mo 
class="MathClass-punc">,</mo><mi 
>r</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">.</mo></mtd>                            
<mtd></mtd>                              </mtr></mtable>
</math>
<!--l. 2070--><p class="nopar">
Some text after to test the below-display spacing.
                                                                  

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{gather*}
&#x00A0;<br />D(a,r)&#x005C;equiv&#x005C;{z&#x005C;in&#x005C;mathbf{C}&#x005C;colon&#x00A0;&#x005C;abs{z-a}&#x003C;r&#x005C;},&#x005C;&#x005C;
&#x00A0;<br />&#x005C;seg&#x00A0;(a,r)&#x005C;equiv&#x005C;{z&#x005C;in&#x005C;mathbf{C}&#x005C;colon
&#x00A0;<br />&#x005C;Im&#x00A0;z=&#x00A0;&#x005C;Im&#x00A0;a,&#x005C;&#x00A0;&#x005C;abs{z-a}&#x003C;r&#x005C;},&#x005C;&#x005C;
&#x00A0;<br />c(e,&#x005C;theta,r)&#x005C;equiv&#x005C;{(x,y)&#x005C;in&#x005C;mathbf{C}
&#x00A0;<br />&#x00A0;&#x005C;colon&#x00A0;&#x005C;abs{x-e}&#x003C;y&#x005C;tan&#x005C;theta,&#x005C;&#x00A0;0&#x003C;y&#x003C;r&#x005C;},&#x005C;&#x005C;
&#x00A0;<br />C(E,&#x005C;theta,r)&#x005C;equiv&#x005C;bigcup&#x005F;{e&#x005C;in&#x00A0;E}c(e,&#x005C;theta,r).
&#x00A0;<br />&#x005C;end{gather*}
</div>
</td></tr></table>
                                                                  

   <h4 class="subsectionHead"><span class="titlemark">A.4</span> <a 
  name="x1-37000A.4"></a>Align</h4> Numbered version:
<!--tex4ht:inline--><!--l. 2092--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
display="block"><mtable 
class="align">
             <mtr><mtd 
class="align-odd"><msub><mrow 
><mi 
>&#x03B3;</mi></mrow><mrow 
><mi 
>x</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd>             <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
>cos</mo><!--nolimits--><mi 
>t</mi><mi 
>u</mi> <mo 
class="MathClass-bin">+</mo><mo 
> sin</mo><!--nolimits--><mi 
>t</mi><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>v</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo> </mtd>                            <mtd 
class="align-label">             <mstyle 
    class="label" id="x1-37001r72"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>7</mn><mn>2</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
             </mtd></mtr><mtr><mtd 
class="align-odd"><msub><mrow 
><mi 
>&#x03B3;</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd>             <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mo 
>cos</mo><!--nolimits--><mi 
>t</mi><mi 
>v</mi> <mo 
class="MathClass-bin">+</mo><mo 
> sin</mo><!--nolimits--><mi 
>t</mi><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo> </mtd>                            <mtd 
class="align-label">             <mstyle 
    class="label" id="x1-37002r73"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>7</mn><mn>3</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
             </mtd></mtr><mtr><mtd 
class="align-odd"><msub><mrow 
><mi 
>&#x03B3;</mi></mrow><mrow 
><mi 
>z</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd>             <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="("  close=")" ><mo 
>cos</mo><!--nolimits--><mi 
>t</mi><mi 
>u</mi> <mo 
class="MathClass-bin">+</mo> <mfrac><mrow 
><mi 
>&#x03B1;</mi></mrow> 
<mrow 
><mi 
>&#x03B2;</mi></mrow></mfrac><mo 
>sin</mo><!--nolimits--><mi 
>t</mi><mi 
>v</mi><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-bin">&#x2212;</mo><mfrac><mrow 
><mi 
>&#x03B2;</mi></mrow>
<mrow 
><mi 
>&#x03B1;</mi></mrow></mfrac><mo 
>sin</mo><!--nolimits--><mi 
>t</mi><mi 
>u</mi> <mo 
class="MathClass-bin">+</mo><mo 
> cos</mo><!--nolimits--><mi 
>t</mi><mi 
>v</mi></mfenced><mo 
class="MathClass-punc">.</mo> </mtd>             <mtd 
class="align-label">             <mstyle 
    class="label" id="x1-37003r74"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>7</mn><mn>4</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
   </mtd></mtr></mtable></math>
Some text after to test the below-display spacing.
                                                                  

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{align}
&#x00A0;<br />&#x005C;gamma&#x005F;x(t)&#x0026;=(&#x005C;cos&#x00A0;tu+&#x005C;sin&#x00A0;tx,v),&#x005C;&#x005C;
&#x00A0;<br />&#x005C;gamma&#x005F;y(t)&#x0026;=(u,&#x005C;cos&#x00A0;tv+&#x005C;sin&#x00A0;ty),&#x005C;&#x005C;
&#x00A0;<br />&#x005C;gamma&#x005F;z(t)&#x0026;=&#x005C;left(&#x005C;cos&#x00A0;tu+&#x005C;frac&#x005C;alpha&#x005C;beta&#x005C;sin&#x00A0;tv,
&#x00A0;<br />&#x00A0;&#x00A0;-&#x005C;frac&#x005C;beta&#x005C;alpha&#x005C;sin&#x00A0;tu+&#x005C;cos&#x00A0;tv&#x005C;right).
&#x00A0;<br />&#x005C;end{align}
</div>
</td></tr></table>
<!--l. 2105--><p class="indent">   Unnumbered version:
<!--tex4ht:inline--></p><!--l. 2111--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
display="block"><mtable 
class="align-star">
             <mtr><mtd 
class="align-odd"><msub><mrow 
><mi 
>&#x03B3;</mi></mrow><mrow 
><mi 
>x</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd>             <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><mo 
>cos</mo><!--nolimits--><mi 
>t</mi><mi 
>u</mi> <mo 
class="MathClass-bin">+</mo><mo 
> sin</mo><!--nolimits--><mi 
>t</mi><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>v</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo> </mtd>
             </mtr><mtr><mtd 
class="align-odd"><msub><mrow 
><mi 
>&#x03B3;</mi></mrow><mrow 
><mi 
>y</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd>             <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>u</mi><mo 
class="MathClass-punc">,</mo><mo 
>cos</mo><!--nolimits--><mi 
>t</mi><mi 
>v</mi> <mo 
class="MathClass-bin">+</mo><mo 
> sin</mo><!--nolimits--><mi 
>t</mi><mi 
>y</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mo 
class="MathClass-punc">,</mo> </mtd>
             </mtr><mtr><mtd 
class="align-odd"><msub><mrow 
><mi 
>&#x03B3;</mi></mrow><mrow 
><mi 
>z</mi></mrow></msub 
><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>t</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd>             <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <mfenced separators="" 
open="("  close=")" ><mo 
>cos</mo><!--nolimits--><mi 
>t</mi><mi 
>u</mi> <mo 
class="MathClass-bin">+</mo> <mfrac><mrow 
><mi 
>&#x03B1;</mi></mrow> 
<mrow 
><mi 
>&#x03B2;</mi></mrow></mfrac><mo 
>sin</mo><!--nolimits--><mi 
>t</mi><mi 
>v</mi><mo 
class="MathClass-punc">,</mo><mo 
class="MathClass-bin">&#x2212;</mo><mfrac><mrow 
><mi 
>&#x03B2;</mi></mrow>
<mrow 
><mi 
>&#x03B1;</mi></mrow></mfrac><mo 
>sin</mo><!--nolimits--><mi 
>t</mi><mi 
>u</mi> <mo 
class="MathClass-bin">+</mo><mo 
> cos</mo><!--nolimits--><mi 
>t</mi><mi 
>v</mi></mfenced><mo 
class="MathClass-punc">.</mo> </mtd>
   </mtr></mtable></math>
Some text after to test the below-display spacing.
                                                                  

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{align*}
&#x00A0;<br />&#x005C;gamma&#x005F;x(t)&#x0026;=(&#x005C;cos&#x00A0;tu+&#x005C;sin&#x00A0;tx,v),&#x005C;&#x005C;
&#x00A0;<br />&#x005C;gamma&#x005F;y(t)&#x0026;=(u,&#x005C;cos&#x00A0;tv+&#x005C;sin&#x00A0;ty),&#x005C;&#x005C;
&#x00A0;<br />&#x005C;gamma&#x005F;z(t)&#x0026;=&#x005C;left(&#x005C;cos&#x00A0;tu+&#x005C;frac&#x005C;alpha&#x005C;beta&#x005C;sin&#x00A0;tv,
&#x00A0;<br />&#x00A0;&#x00A0;-&#x005C;frac&#x005C;beta&#x005C;alpha&#x005C;sin&#x00A0;tu+&#x005C;cos&#x00A0;tv&#x005C;right).
&#x00A0;<br />&#x005C;end{align*}
</div>
</td></tr></table>
<!--l. 2124--><p class="indent">   A variation:
<!--tex4ht:inline--></p><!--l. 2129--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
display="block"><mtable 
class="align">
              <mtr><mtd 
class="align-odd"><mi 
>x</mi></mtd>                  <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <mi 
>y</mi></mtd>                  <mtd 
class="align-odd"></mtd>              <mtd 
class="align-even"><!--mstyle 
class="text"--><mtext >by&#x000A0;(</mtext><mtext 
 xref="x1-39003r84"  class="label" >84<!--tex4ht:ref: eq:C--></mtext><mtext 
class="endlabel">)</mtext><!--/mstyle--></mtd>                                              <mtd 
class="align-label">                 <mstyle 
    class="label" id="x1-37004r75"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>7</mn><mn>5</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
              </mtd></mtr><mtr><mtd 
class="align-odd"><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mtd>                  <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mtd>                  <mtd 
class="align-odd"></mtd>              <mtd 
class="align-even"><!--mstyle 
class="text"--><mtext >by&#x000A0;(</mtext><mtext 
 xref="x1-39004r85"  class="label" >85<!--tex4ht:ref: eq:D--></mtext><mtext 
class="endlabel">)</mtext><!--/mstyle--></mtd>                                              <mtd 
class="align-label">                 <mstyle 
    class="label" id="x1-37005r76"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>7</mn><mn>6</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
              </mtd></mtr><mtr><mtd 
class="align-odd"><mi 
>x</mi> <mo 
class="MathClass-bin">+</mo> <msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mtd>              <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mtd>              <mtd 
class="align-odd"></mtd>              <mtd 
class="align-even"><!--mstyle 
class="text"--><mtext >by&#x000A0;Axiom&#x000A0;1.</mtext><!--/mstyle--></mtd>                                          <mtd 
class="align-label">                 <mstyle 
    class="label" id="x1-37006r77"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>7</mn><mn>7</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
   </mtd></mtr></mtable></math>
Some text after to test the below-display spacing.
                                                                  

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{align}
&#x00A0;<br />x&#x0026;&#x00A0;=y&#x00A0;&#x0026;&#x0026;&#x00A0;&#x005C;text&#x00A0;{by&#x00A0;(&#x005C;ref{eq:C})}&#x005C;&#x005C;
&#x00A0;<br />x&#x2019;&#x0026;&#x00A0;=&#x00A0;y&#x2019;&#x00A0;&#x0026;&#x0026;&#x00A0;&#x005C;text&#x00A0;{by&#x00A0;(&#x005C;ref{eq:D})}&#x005C;&#x005C;
&#x00A0;<br />x+x&#x2019;&#x00A0;&#x0026;&#x00A0;=&#x00A0;y+y&#x2019;&#x00A0;&#x0026;&#x0026;&#x00A0;&#x005C;text&#x00A0;{by&#x00A0;Axiom&#x00A0;1.}
&#x00A0;<br />&#x005C;end{align}
</div>
</td></tr></table>
                                                                  

   <h4 class="subsectionHead"><span class="titlemark">A.5</span> <a 
  name="x1-38000A.5"></a>Align and split within gather</h4> When using the <span 
class="cmtt-10">align </span>environment within the
<span 
class="cmtt-10">gather </span>environment, one or the other, or both, should be unnumbered (using the
<span class="obeylines-h"><span 
class="cmtt-10">*</span></span> form); numbering both the outer and inner environment would cause a
conflict.
<!--l. 2148--><p class="noindent">Automatically numbered <span 
class="cmtt-10">gather </span>with <span 
class="cmtt-10">split </span>and <span 
class="cmtt-10">align*</span>:
</p><!--l. 2149--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     
<mtable 
class="gather">
<mtr> 
<mtd>    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow> <mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>z</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>&#x03B3;</mi></mrow><mrow 
><mn>1</mn><mn>0</mn></mrow></msub 
><mi 
>x</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>&#x03B3;</mi></mrow><mrow 
><mi 
>m</mi><mi 
>n</mi></mrow></msub 
><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>m</mi></mrow></msup 
><msup><mrow 
><mi 
>z</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>          <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>z</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>M</mi><msup><mrow 
><mi 
>r</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msup 
><mi 
>x</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>M</mi><msup><mrow 
><mi 
>r</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>m</mi><mo 
class="MathClass-bin">+</mo><mi 
>n</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>m</mi></mrow></msup 
><msup><mrow 
><mi 
>z</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
>
     </mtd></mtr></mtable>                                                                                                     </mtd> 
<mtd><mstyle 
    class="label" id="x1-38001r78"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>7</mn><mn>8</mn></mrow><mo 
class="MathClass-close">)</mo></mrow></mtd>
</mtr><mtr> 
<mtd><mrow>
    <mtable 
class="align-star">
                                            <mtr><mtd 
class="align-odd"><msup><mrow 
><mi 
>&#x03B6;</mi></mrow><mrow 
><mn>0</mn></mrow></msup 
></mtd>                                                       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>&#x03BE;</mi></mrow><mrow 
><mn>0</mn></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
><mo 
class="MathClass-punc">,</mo></mtd>
                                      </mtr><mtr><mtd 
class="align-odd"><msup><mrow 
><mi 
>&#x03B6;</mi></mrow><mrow 
><mn>1</mn></mrow></msup 
></mtd>                                                       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msup><mrow 
><mi 
>&#x03BE;</mi></mrow><mrow 
><mn>0</mn></mrow></msup 
><msup><mrow 
><mi 
>&#x03BE;</mi></mrow><mrow 
><mn>1</mn></mrow></msup 
><mo 
class="MathClass-punc">,</mo></mtd>
                                      </mtr><mtr><mtd 
class="align-odd"><msup><mrow 
><mi 
>&#x03B6;</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
></mtd>                                                       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>&#x03BE;</mi></mrow><mrow 
><mn>1</mn></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
><mo 
class="MathClass-punc">,</mo></mtd>
    </mtr></mtable></mrow>                                                                                    </mtd> 
<mtd></mtd>     </mtr></mtable>
</math>
<!--l. 2159--><p class="nopar">
Here the <span 
class="cmtt-10">split </span>environment gets a number from the outer <span 
class="cmtt-10">gather </span>environment;
numbers for individual lines of the <span 
class="cmtt-10">align* </span>are suppressed because of the
star.
                                                                  

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{gather}
&#x00A0;<br />&#x005C;begin{split}&#x00A0;&#x005C;varphi(x,z)
&#x00A0;<br />&#x0026;=z-&#x005C;gamma&#x005F;{10}x-&#x005C;gamma&#x005F;{mn}x&#x02C6;mz&#x02C6;n&#x005C;&#x005C;
&#x00A0;<br />&#x0026;=z-Mr&#x02C6;{-1}x-Mr&#x02C6;{-(m+n)}x&#x02C6;mz&#x02C6;n
&#x00A0;<br />&#x005C;end{split}&#x005C;&#x005C;[6pt]
&#x00A0;<br />&#x005C;begin{align*}
&#x00A0;<br />&#x005C;zeta&#x02C6;0&#x00A0;&#x0026;=(&#x005C;xi&#x02C6;0)&#x02C6;2,&#x005C;&#x005C;
&#x00A0;<br />&#x005C;zeta&#x02C6;1&#x00A0;&#x0026;=&#x005C;xi&#x02C6;0&#x005C;xi&#x02C6;1,&#x005C;&#x005C;
&#x00A0;<br />&#x005C;zeta&#x02C6;2&#x00A0;&#x0026;=(&#x005C;xi&#x02C6;1)&#x02C6;2,
&#x00A0;<br />&#x005C;end{align*}
&#x00A0;<br />&#x005C;end{gather}
</div>
</td></tr></table>
<!--l. 2179--><p class="indent">   The <span class="obeylines-h"><span 
class="cmtt-10">*</span></span>-ed form of <span 
class="cmtt-10">gather </span>with the non-<span class="obeylines-h"><span 
class="cmtt-10">*</span></span>-ed form of <span 
class="cmtt-10">align</span>.
</p><!--l. 2181--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
mode="display">     
<mtable 
class="gather-star">
<mtr> 
<mtd>    <mtable class="split"><mtr><mtd>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow> <mi 
>&#x03D5;</mi><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi><mo 
class="MathClass-punc">,</mo><mi 
>z</mi></mrow><mo 
class="MathClass-close">)</mo></mrow><mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>z</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>&#x03B3;</mi></mrow><mrow 
><mn>1</mn><mn>0</mn></mrow></msub 
><mi 
>x</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>&#x03B3;</mi></mrow><mrow 
><mi 
>m</mi><mi 
>n</mi></mrow></msub 
><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>m</mi></mrow></msup 
><msup><mrow 
><mi 
>z</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
>
<mrow 
class="split-mtr"></mrow><mrow 
class="split-mtd"></mrow>          <mrow 
class="split-mtd"></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>z</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>M</mi><msup><mrow 
><mi 
>r</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mn>1</mn></mrow></msup 
><mi 
>x</mi> <mo 
class="MathClass-bin">&#x2212;</mo> <mi 
>M</mi><msup><mrow 
><mi 
>r</mi></mrow><mrow 
><mo 
class="MathClass-bin">&#x2212;</mo><mrow><mo 
class="MathClass-open">(</mo><mrow><mi 
>m</mi><mo 
class="MathClass-bin">+</mo><mi 
>n</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow></msup 
><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>m</mi></mrow></msup 
><msup><mrow 
><mi 
>z</mi></mrow><mrow 
><mi 
>n</mi></mrow></msup 
>
     </mtd></mtr></mtable>                                                                                                     </mtd>                                                                                                                                         
<mtd></mtd>
</mtr><mtr> 
<mtd><mrow>
     <mtable 
class="align">
                                             <mtr><mtd 
class="align-odd"><msup><mrow 
><mi 
>&#x03B6;</mi></mrow><mrow 
><mn>0</mn></mrow></msup 
></mtd>                                                       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>&#x03BE;</mi></mrow><mrow 
><mn>0</mn></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
><mo 
class="MathClass-punc">,</mo></mtd>                                      <mtd 
class="align-label">                                 <mstyle 
    class="label" id="x1-38002r79"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>7</mn><mn>9</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
                                            </mtd></mtr><mtr><mtd 
class="align-odd"><msup><mrow 
><mi 
>&#x03B6;</mi></mrow><mrow 
><mn>1</mn></mrow></msup 
></mtd>                                                       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msup><mrow 
><mi 
>&#x03BE;</mi></mrow><mrow 
><mn>0</mn></mrow></msup 
><msup><mrow 
><mi 
>&#x03BE;</mi></mrow><mrow 
><mn>1</mn></mrow></msup 
><mo 
class="MathClass-punc">,</mo></mtd>                                       <mtd 
class="align-label">                                 <mstyle 
    class="label" id="x1-38003r80"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>8</mn><mn>0</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
                                            </mtd></mtr><mtr><mtd 
class="align-odd"><msup><mrow 
><mi 
>&#x03B6;</mi></mrow><mrow 
><mn>2</mn></mrow></msup 
></mtd>                                                       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msup><mrow 
><mrow><mo 
class="MathClass-open">(</mo><mrow><msup><mrow 
><mi 
>&#x03BE;</mi></mrow><mrow 
><mn>1</mn></mrow></msup 
></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mrow 
><mn>2</mn></mrow></msup 
><mo 
class="MathClass-punc">,</mo></mtd>                                      <mtd 
class="align-label">                                 <mstyle 
    class="label" id="x1-38004r81"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>8</mn><mn>1</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
    </mtd></mtr></mtable></mrow>                                                                                                                                                                                                                                        </mtd> 
<mtd></mtd></mtr></mtable>
</math>

<!--l. 2190--><p class="nopar">
Some text after to test the below-display spacing.
                                                                  

</p>
   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{gather*}
&#x00A0;<br />&#x005C;begin{split}&#x00A0;&#x005C;varphi(x,z)
&#x00A0;<br />&#x0026;=z-&#x005C;gamma&#x005F;{10}x-&#x005C;gamma&#x005F;{mn}x&#x02C6;mz&#x02C6;n&#x005C;&#x005C;
&#x00A0;<br />&#x0026;=z-Mr&#x02C6;{-1}x-Mr&#x02C6;{-(m+n)}x&#x02C6;mz&#x02C6;n
&#x00A0;<br />&#x005C;end{split}&#x005C;&#x005C;[6pt]
&#x00A0;<br />&#x005C;begin{align}&#x00A0;&#x005C;zeta&#x02C6;0&#x0026;=(&#x005C;xi&#x02C6;0)&#x02C6;2,&#x005C;&#x005C;
&#x00A0;<br />&#x005C;zeta&#x02C6;1&#x00A0;&#x0026;=&#x005C;xi&#x02C6;0&#x005C;xi&#x02C6;1,&#x005C;&#x005C;
&#x00A0;<br />&#x005C;zeta&#x02C6;2&#x00A0;&#x0026;=(&#x005C;xi&#x02C6;1)&#x02C6;2,
&#x00A0;<br />&#x005C;end{align}
&#x00A0;<br />&#x005C;end{gather*}
</div>
</td></tr></table>
                                                                  

   <h4 class="subsectionHead"><span class="titlemark">A.6</span> <a 
  name="x1-39000A.6"></a>Alignat</h4> Numbered version:
<!--tex4ht:inline--><!--l. 2216--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
display="block"><mtable 
class="alignat">
       <mtr><mtd 
class="align-odd"><msub><mrow 
><mi 
>V</mi> </mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mtd>       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>q</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><msub><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo></mtd>       <mtd 
class="align-odd"><mspace width="2em" class="qquad"/><msub><mrow 
><mi 
>X</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mtd>       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>q</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><msub><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo></mtd>       <mtd 
class="align-odd"><mspace width="2em" class="qquad"/><msub><mrow 
><mi 
>U</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mtd>       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mspace width="2em" class="qquad"/><!--mstyle 
class="text"--><mtext >for&#x000A0;</mtext><!--mstyle 
class="math"--><mi 
>i</mi><mo 
class="MathClass-rel">&#x2260;</mo><mi 
>j</mi><!--/mstyle--><mtext >;</mtext><!--/mstyle--></mtd>                                            <mtd 
class="align-label">      <mstyle 
    class="label" id="x1-39001r82"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>8</mn><mn>2</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
       </mtd></mtr><mtr><mtd 
class="align-odd"><msub><mrow 
><mi 
>V</mi> </mrow><mrow 
><mi 
>j</mi></mrow></msub 
></mtd>       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo></mtd>            <mtd 
class="align-odd"><mspace width="2em" class="qquad"/><msub><mrow 
><mi 
>X</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
></mtd>       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo></mtd>             <mtd 
class="align-odd"><mspace width="2em" class="qquad"/><msub><mrow 
><mi 
>U</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
></mtd>       <mtd 
class="align-even"><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-bin">+</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2260;</mo><mi 
>j</mi></mrow></msub 
><msub><mrow 
><mi 
>q</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mo 
class="MathClass-punc">.</mo></mtd>                                          <mtd 
class="align-label">     <mstyle 
    class="label" id="x1-39002r83"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>8</mn><mn>3</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
   </mtd></mtr></mtable></math>
Some text after to test the below-display spacing.
                                                                  

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{alignat}{3}
&#x00A0;<br />V&#x005F;i&#x00A0;&#x0026;&#x00A0;=v&#x005F;i&#x00A0;-&#x00A0;q&#x005F;i&#x00A0;v&#x005F;j,&#x00A0;&#x0026;&#x00A0;&#x005C;qquad&#x00A0;X&#x005F;i&#x00A0;&#x0026;&#x00A0;=&#x00A0;x&#x005F;i&#x00A0;-&#x00A0;q&#x005F;i&#x00A0;x&#x005F;j,
&#x00A0;<br />&#x00A0;&#x0026;&#x00A0;&#x005C;qquad&#x00A0;U&#x005F;i&#x00A0;&#x0026;&#x00A0;=&#x00A0;u&#x005F;i,
&#x00A0;<br />&#x00A0;&#x005C;qquad&#x00A0;&#x005C;text{for&#x00A0;$i&#x005C;ne&#x00A0;j$;}&#x005C;label{eq:B}&#x005C;&#x005C;
&#x00A0;<br />V&#x005F;j&#x00A0;&#x0026;&#x00A0;=&#x00A0;v&#x005F;j,&#x00A0;&#x0026;&#x00A0;&#x005C;qquad&#x00A0;X&#x005F;j&#x00A0;&#x0026;&#x00A0;=&#x00A0;x&#x005F;j,
&#x00A0;<br />&#x00A0;&#x00A0;&#x0026;&#x00A0;&#x005C;qquad&#x00A0;U&#x005F;j&#x00A0;&#x0026;&#x00A0;u&#x005F;j&#x00A0;+&#x00A0;&#x005C;sum&#x005F;{i&#x005C;ne&#x00A0;j}&#x00A0;q&#x005F;i&#x00A0;u&#x005F;i.
&#x00A0;<br />&#x005C;end{alignat}
</div>
</td></tr></table>
<!--l. 2230--><p class="indent">   Unnumbered version:
<!--tex4ht:inline--></p><!--l. 2237--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
display="block"><mtable 
class="alignat-star">
       <mtr><mtd 
class="align-odd"><msub><mrow 
><mi 
>V</mi> </mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mtd>       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>q</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><msub><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo></mtd>       <mtd 
class="align-odd"><mspace width="2em" class="qquad"/><msub><mrow 
><mi 
>X</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mtd>       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
> <mo 
class="MathClass-bin">&#x2212;</mo> <msub><mrow 
><mi 
>q</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><msub><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo></mtd>       <mtd 
class="align-odd"><mspace width="2em" class="qquad"/><msub><mrow 
><mi 
>U</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
></mtd>       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo><mspace width="2em" class="qquad"/><!--mstyle 
class="text"--><mtext >for&#x000A0;</mtext><!--mstyle 
class="math"--><mi 
>i</mi><mo 
class="MathClass-rel">&#x2260;</mo><mi 
>j</mi><!--/mstyle--><mtext >;</mtext><!--/mstyle--></mtd>
       </mtr><mtr><mtd 
class="align-odd"><msub><mrow 
><mi 
>V</mi> </mrow><mrow 
><mi 
>j</mi></mrow></msub 
></mtd>       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>v</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo></mtd>            <mtd 
class="align-odd"><mspace width="2em" class="qquad"/><msub><mrow 
><mi 
>X</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
></mtd>       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msub><mrow 
><mi 
>x</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
><mo 
class="MathClass-punc">,</mo></mtd>             <mtd 
class="align-odd"><mspace width="2em" class="qquad"/><msub><mrow 
><mi 
>U</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
></mtd>       <mtd 
class="align-even"><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>j</mi></mrow></msub 
> <mo 
class="MathClass-bin">+</mo><msub><mrow 
> <mo 
class="MathClass-op">&#x2211;</mo>
   </mrow><mrow 
><mi 
>i</mi><mo 
class="MathClass-rel">&#x2260;</mo><mi 
>j</mi></mrow></msub 
><msub><mrow 
><mi 
>q</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><msub><mrow 
><mi 
>u</mi></mrow><mrow 
><mi 
>i</mi></mrow></msub 
><mo 
class="MathClass-punc">.</mo></mtd>
   </mtr></mtable></math>
Some text after to test the below-display spacing.
                                                                  

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{alignat*}3
&#x00A0;<br />V&#x005F;i&#x00A0;&#x0026;&#x00A0;=v&#x005F;i&#x00A0;-&#x00A0;q&#x005F;i&#x00A0;v&#x005F;j,&#x00A0;&#x0026;&#x00A0;&#x005C;qquad&#x00A0;X&#x005F;i&#x00A0;&#x0026;&#x00A0;=&#x00A0;x&#x005F;i&#x00A0;-&#x00A0;q&#x005F;i&#x00A0;x&#x005F;j,
&#x00A0;<br />&#x00A0;&#x0026;&#x00A0;&#x005C;qquad&#x00A0;U&#x005F;i&#x00A0;&#x0026;&#x00A0;=&#x00A0;u&#x005F;i,
&#x00A0;<br />&#x00A0;&#x005C;qquad&#x00A0;&#x005C;text{for&#x00A0;$i&#x005C;ne&#x00A0;j$;}&#x00A0;&#x005C;&#x005C;
&#x00A0;<br />V&#x005F;j&#x00A0;&#x0026;&#x00A0;=&#x00A0;v&#x005F;j,&#x00A0;&#x0026;&#x00A0;&#x005C;qquad&#x00A0;X&#x005F;j&#x00A0;&#x0026;&#x00A0;=&#x00A0;x&#x005F;j,
&#x00A0;<br />&#x00A0;&#x00A0;&#x0026;&#x00A0;&#x005C;qquad&#x00A0;U&#x005F;j&#x00A0;&#x0026;&#x00A0;u&#x005F;j&#x00A0;+&#x00A0;&#x005C;sum&#x005F;{i&#x005C;ne&#x00A0;j}&#x00A0;q&#x005F;i&#x00A0;u&#x005F;i.
&#x00A0;<br />&#x005C;end{alignat*}
</div>
</td></tr></table>
                                                                  

<!--l. 2252--><p class="indent">   The most common use for <span 
class="cmtt-10">alignat </span>is for things like
<!--tex4ht:inline--></p><!--l. 2257--><math 
xmlns="http://www.w3.org/1998/Math/MathML" 
display="block"><mtable 
class="alignat">
                    <mtr><mtd 
class="align-odd"><mi 
>x</mi></mtd>                        <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <mi 
>y</mi></mtd>                        <mtd 
class="align-odd"></mtd>                    <mtd 
class="align-even"><mspace width="2em" class="qquad"/><!--mstyle 
class="text"--><mtext >by&#x000A0;(</mtext><mtext 
 xref="x1-34003r66"  class="label" >66<!--tex4ht:ref: eq:A--></mtext><mtext 
class="endlabel">)</mtext><!--/mstyle--></mtd>                                                                                    <mtd 
class="align-label">                          <mstyle 
    class="label" id="x1-39003r84"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>8</mn><mn>4</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
                    </mtd></mtr><mtr><mtd 
class="align-odd"><msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mtd>                       <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mtd>                       <mtd 
class="align-odd"></mtd>                    <mtd 
class="align-even"><mspace width="2em" class="qquad"/><!--mstyle 
class="text"--><mtext >by&#x000A0;(</mtext><mtext 
 xref="x1-39001r82"  class="label" >82<!--tex4ht:ref: eq:B--></mtext><mtext 
class="endlabel">)</mtext><!--/mstyle--></mtd>                                                                                    <mtd 
class="align-label">                          <mstyle 
    class="label" id="x1-39004r85"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>8</mn><mn>5</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
                    </mtd></mtr><mtr><mtd 
class="align-odd"><mi 
>x</mi> <mo 
class="MathClass-bin">+</mo> <msup><mrow 
><mi 
>x</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mtd>                    <mtd 
class="align-even"> <mo 
class="MathClass-rel">=</mo> <mi 
>y</mi> <mo 
class="MathClass-bin">+</mo> <msup><mrow 
><mi 
>y</mi></mrow><mrow 
><mo 
class="MathClass-op">&#x2032;</mo></mrow></msup 
></mtd>                    <mtd 
class="align-odd"></mtd>                    <mtd 
class="align-even"><mspace width="2em" class="qquad"/><!--mstyle 
class="text"--><mtext >by&#x000A0;Axiom&#x000A0;1.</mtext><!--/mstyle--></mtd>                                                                                <mtd 
class="align-label">                          <mstyle 
    class="label" id="x1-39005r86"  ></mstyle><!--endlabel--><mrow><mo 
class="MathClass-open">(</mo><mrow><mn>8</mn><mn>6</mn></mrow><mo 
class="MathClass-close">)</mo></mrow>
   </mtd></mtr></mtable></math>
Some text after to test the below-display spacing.
                                                                  

   <table width="100%" 
class="verbatim"><tr class="verbatim"><td 
class="verbatim"><div class="verbatim">
&#x005C;begin{alignat}{2}
&#x00A0;<br />x&#x0026;&#x00A0;=y&#x00A0;&#x0026;&#x0026;&#x00A0;&#x005C;qquad&#x00A0;&#x005C;text&#x00A0;{by&#x00A0;(&#x005C;ref{eq:A})}&#x005C;label{eq:C}&#x005C;&#x005C;
&#x00A0;<br />x&#x2019;&#x0026;&#x00A0;=&#x00A0;y&#x2019;&#x00A0;&#x0026;&#x0026;&#x00A0;&#x005C;qquad&#x00A0;&#x005C;text&#x00A0;{by&#x00A0;(&#x005C;ref{eq:B})}&#x005C;label{eq:D}&#x005C;&#x005C;
&#x00A0;<br />x+x&#x2019;&#x00A0;&#x0026;&#x00A0;=&#x00A0;y+y&#x2019;&#x00A0;&#x0026;&#x0026;&#x00A0;&#x005C;qquad&#x00A0;&#x005C;text&#x00A0;{by&#x00A0;Axiom&#x00A0;1.}
&#x00A0;<br />&#x005C;end{alignat}
</div>
</td></tr></table>
                                                                  

   <h3 class="likesectionHead"><a 
  name="x1-40000A.6"></a>References</h3>
    <div class="thebibliography"><p class="bibitem"><span class="biblabel">
  [1]<span class="bibsp">&#x00A0;&#x00A0;&#x00A0;</span></span><a 
  name="Xdihe:newdir"></a>W.&#x00A0;Diffie   and   E.&#x00A0;Hellman,   <span 
class="cmti-10">New   directions   in   cryptography</span>,   IEEE
    Transactions on Information Theory <span 
class="cmbx-10">22 </span>(1976), no.&#x00A0;5, 644-654.
    </p><p class="bibitem"><span class="biblabel">
  [2]<span class="bibsp">&#x00A0;&#x00A0;&#x00A0;</span></span><a 
  name="Xfre:cichon"></a>D.&#x00A0;H. Fremlin, <span 
class="cmti-10">Cichon&#x2019;s diagram</span>, 1983/1984, presented at the S&#x00E9;minaire
    Initiation &#x00E0; l&#x2019;Analyse, G. Choquet, M. Rogalski, J. Saint Raymond, at the
    Universit&#x00E9; Pierre et Marie Curie, Paris, 23e ann&#x00E9;e.
    </p><p class="bibitem"><span class="biblabel">
  [3]<span class="bibsp">&#x00A0;&#x00A0;&#x00A0;</span></span><a 
  name="Xgouja:lagrmeth"></a>I.&#x00A0;P. Goulden and D.&#x00A0;M. Jackson, <span 
class="cmti-10">The enumeration of directed closed Euler</span>
    <span 
class="cmti-10">trails and directed Hamiltonian circuits by Langrangian methods</span>, European
    J. Combin. <span 
class="cmbx-10">2 </span>(1981), 131-212.
    </p><p class="bibitem"><span class="biblabel">
  [4]<span class="bibsp">&#x00A0;&#x00A0;&#x00A0;</span></span><a 
  name="Xhapa:graphenum"></a>F.&#x00A0;Harary and E.&#x00A0;M. Palmer, <span 
class="cmti-10">Graphical enumeration</span>, Academic Press, 1973.
    </p><p class="bibitem"><span class="biblabel">
  [5]<span class="bibsp">&#x00A0;&#x00A0;&#x00A0;</span></span><a 
  name="Ximlelu:oneway"></a>R.&#x00A0;Impagliazzo,  L.&#x00A0;Levin,  and  M.&#x00A0;Luby,  <span 
class="cmti-10">Pseudo-random  generation  from</span>
    <span 
class="cmti-10">one-way functions</span>, Proc. 21st STOC (1989), ACM, New York, pp.&#x00A0;12-24.
    </p><p class="bibitem"><span class="biblabel">
  [6]<span class="bibsp">&#x00A0;&#x00A0;&#x00A0;</span></span><a 
  name="Xkomiyo:unipfunc"></a>M.&#x00A0;Kojima,  S.&#x00A0;Mizuno,  and  A.&#x00A0;Yoshise,  <span 
class="cmti-10">A  new  continuation  method  for</span>
    <span 
class="cmti-10">complementarity problems with uniform p-functions</span>, Tech. Report B-194,
    Tokyo Inst. of Technology, Tokyo, 1987, Dept. of Information Sciences.
    </p><p class="bibitem"><span class="biblabel">
  [7]<span class="bibsp">&#x00A0;&#x00A0;&#x00A0;</span></span><a 
  name="Xkomiyo:lincomp"></a>______, <span 
class="cmti-10">A polynomial-time algorithm for a class of linear complementarity</span>
    <span 
class="cmti-10">problems</span>, Tech. Report B-193, Tokyo Inst. of Technology, Tokyo, 1987,
    Dept. of Information Sciences.
    </p><p class="bibitem"><span class="biblabel">
  [8]<span class="bibsp">&#x00A0;&#x00A0;&#x00A0;</span></span><a 
  name="Xliuchow:formalsum"></a>C.&#x00A0;J. Liu and Yutze Chow, <span 
class="cmti-10">On operator and formal sum methods for graph</span>
    <span 
class="cmti-10">enumeration problems</span>, SIAM J. Algorithms Discrete Methods <span 
class="cmbx-10">5 </span>(1984),
    384-438.
    </p><p class="bibitem"><span class="biblabel">
  [9]<span class="bibsp">&#x00A0;&#x00A0;&#x00A0;</span></span><a 
  name="Xmami:matrixth"></a>M.&#x00A0;Marcus and H.&#x00A0;Minc, <span 
class="cmti-10">A survey of matrix theory and matrix inequalities</span>,
    Complementary Series in Math. <span 
class="cmbx-10">14 </span>(1964), 21-48.
    </p><p class="bibitem"><span class="biblabel">
 [10]<span class="bibsp">&#x00A0;&#x00A0;&#x00A0;</span></span><a 
  name="Xmiyoki:lincomp"></a>S.&#x00A0;Mizuno, A.&#x00A0;Yoshise, and T.&#x00A0;Kikuchi, <span 
class="cmti-10">Practical polynomial time algorithms</span>
    <span 
class="cmti-10">for  linear  complementarity  problems</span>,  Tech.  Report&#x00A0;13,  Tokyo  Inst.  of
    Technology,  Tokyo,  April  1988,  Dept.  of  Industrial  Engineering  and
    Management.

    </p><p class="bibitem"><span class="biblabel">
 [11]<span class="bibsp">&#x00A0;&#x00A0;&#x00A0;</span></span><a 
  name="Xmoad:quadpro"></a>R.&#x00A0;D. Monteiro and I.&#x00A0;Adler, <span 
class="cmti-10">Interior path following primal-dual algorithms,</span>
    <span 
class="cmti-10">part II: Quadratic programming</span>, August 1987, Working paper, Dept. of
    Industrial Engineering and Operations Research.
    </p><p class="bibitem"><span class="biblabel">
 [12]<span class="bibsp">&#x00A0;&#x00A0;&#x00A0;</span></span><a 
  name="Xste:sint"></a>E.&#x00A0;M. Stein, <span 
class="cmti-10">Singular integrals and differentiability properties of functions</span>,
    Princeton Univ. Press, Princeton, N.J., 1970.
    </p><p class="bibitem"><span class="biblabel">
 [13]<span class="bibsp">&#x00A0;&#x00A0;&#x00A0;</span></span><a 
  name="Xye:intalg"></a>Y.&#x00A0;Ye,  <span 
class="cmti-10">Interior  algorithms  for  linear,  quadratic  and  linearly  constrained</span>
    <span 
class="cmti-10">convex programming</span>, Ph.D. thesis, Stanford Univ., Palo Alto, Calif., July
    1987, Dept. of Engineering-Economic Systems, unpublished.
    </p></div>
    
</body> 
</html> 



